Introduction
Prime-separated rings
In a commutative ring, every radical (equivalently, semiprime) ideal is an intersection of prime ideals [36 Tag 00E0] . We study the corresponding question for one-sided ideals and, more generally, submodules over noncommutative rings.
All rings below are associative with identity, and all modules are unital left modules. A left ideal I I I semiprime if
a R a ⊆ I ⟹ a ∈ I ( a ∈ R ) , aRa \subseteq I \quad\Longrightarrow\quad a \in I \qquad(a \in R), a R a ⊆ I ⟹ a ∈ I ( a ∈ R ) , and a proper left ideal P P P prime if
a R b ⊆ P ⟹ a ∈ P or b ∈ P ( a , b ∈ R ) . aRb \subseteq P \quad\Longrightarrow\quad a \in P\ \text{or}\ b \in P \qquad(a,b \in R). a R b ⊆ P ⟹ a ∈ P or b ∈ P ( a , b ∈ R ) . These are the one-sided notions studied by Dauns [11 1.3] and Hansen [18 Section 1] . Every prime left ideal is semiprime, as is every intersection of prime left ideals. The converse is the question motivating this paper; see also [10 Sections 1.3 and 3] .
2020 Mathematics Subject Classification. Primary 16N60; Secondary 16D25, 16W10, 16R20, 16S10, 13J30.
Key words and phrases. semiprime left ideal, prime left ideal, prime module, subdirect product, ring with involution, real ideal.
The author acknowledges the financial support from the Slovenian Research and Innovation Agency (ARIS), research core funding No. P1-0222 and research projects No. J1-60011 and No. J1-50002.
Question 1. Is every semiprime left ideal of an associative unital ring an intersection of prime left ideals?
A ring with the property in Question 1 is prime-separated . Prime and semiprime submodules are defined in Definition 2.1; for left ideals these notions agree with those above by Lemma 2.2 (4). A module is prime-separated if every semiprime submodule is an intersection of prime submodules (Definition 2.7 ).
Every module over a commutative ring is prime-separated [7 Theorem A] . Rings in which every left ideal is idempotent [18 Lemma 2.1] and matrix rings over commutative rings [5 Theorem 3] are also prime-separated. Our first result gives two ring-theoretic sufficient conditions for every module to be prime-separated. An intersection over an empty family of submodules of M M M M M M
Theorem A. Let R R R
(i) R R R
(ii) R a R RaR R a R a ∈ R a \in R a ∈ R
For every left R R R M M M N ⊆ M N \subseteq M N ⊆ M M M M R R R R R R R R R
Both hypotheses concern R R R k k k k k k k k k
Real prime-separated *-rings
Let R R R a ↦ a ∗ a \mapsto a^* a ↦ a ∗ I I I ∑ i a i ∗ a i ∈ I + I ∗ \sum_i a_i^*a_i \in I+I^* ∑ i a i ∗ a i ∈ I + I ∗ a i ∈ I a_i \in I a i ∈ I weakly real if ∑ i a i ∗ a i ∈ I \sum_i a_i^*a_i \in I ∑ i a i ∗ a i ∈ I a i ∈ I a_i \in I a i ∈ I a i ∈ R a_i \in R a i ∈ R Proposition 3.12 ); real modules are defined in Definition 3.4 . The ring R R R real , or formally real, if its zero ideal is real, and semireal if − 1 -1 − 1 R \mathbb{R} R real prime-separated if every real left ideal, respectively real submodule, is an intersection of real prime ones (Definition 4.1 ). For two-sided ideals this holds in every ring with involution.
Theorem B.
(1) In every ring with involution, every real two-sided ideal is an intersection of real prime two-sided ideals.
(2) There is a left and right Noetherian domain with involution containing a real left ideal that is not an intersection of real prime left ideals. It also contains a prime left ideal that is weakly real but not real.
(3) Let R R R R R R
Part (1) is Theorem 3.1 . The domain in (2) is the Jordan plane R ⟨ x , y ⟩ / ( y x − x y − x 2 ) \mathbb{R}\langle x,y\rangle/(yx-xy-x^2) R ⟨ x , y ⟩ / ( y x − x y − x 2 ) x ∗ = x x^*=x x ∗ = x y ∗ = − y y^*=-y y ∗ = − y Example 3.3 ); its weakly real but nonreal prime left ideal answers [4 Remark 5] negatively. Part (3) follows from the Artinian-factor criterion in Theorem 3.8 . It applies to noncommutative Dedekind prime rings, to Ore extensions D [ x ; σ , δ ] D[x;\sigma,\delta] D [ x ; σ , δ ] σ \sigma σ ({xref}Remark 3.9`).
In a Noetherian or PI ring with involution, semireality is equivalent both to the existence of a proper real prime two-sided ideal and to the existence of a nonzero real module. Theorem 3.11 proves these equivalences whenever R / P R/P R / P P P P
Jacobson properties and PI rings
An affine commutative algebra over a field k k k k k k [36 Tag 00FV] . If k k k [6 Proposition 6.6 and Remark 6.7] ; here and below, real maximal means real and maximal.
A two-sided ideal is primitive if it is the annihilator of a simple module, and real primitive if it is the annihilator of a real simple module. In commutative rings these are the maximal and the real maximal ideals. More generally, in PI rings primitive ideals are precisely maximal two-sided ideals by Kaplansky’s theorem [13 Part B, Theorem 4.13] , and real primitive ideals are precisely real maximal two-sided ideals (Proposition 4.11).
A ring R R R Jacobson if every prime two-sided ideal is an intersection of primitive ideals, and left Jacobson if every prime left ideal is an intersection of maximal left ideals. A left R R R Jacobson if every prime submodule is an intersection of maximal submodules. For rings with involution, the corresponding real versions are obtained by adding the word “real” throughout these definitions. For arbitrary rings, left Jacobson implies Jacobson, and real left Jacobson implies real Jacobson. The converses can fail, but they hold for PI rings.
Theorem C. Let R R R
(1) Every left R R R R R R
(2) R R R R R R
(3) If R R R R R R
(4) If R R R k k k R R R k k k
(5) Suppose that R R R C C C k k k k k k Trd R / C ( a ∗ a ) 1 R \operatorname{Trd}_{R/C}(a^{*}a)1_R Trd R / C ( a ∗ a ) 1 R R R R a ∈ R a \in R a ∈ R R R R
Part (4) is the module form of the Amitsur–Procesi theorem [13 Part B, Theorem 11.7] , which says that affine PI algebras are Jacobson. Part (5) extends to Azumaya algebras the Procesi–Schacher sufficient trace criterion for central simple algebras [32 p. 404] . Real Jacobsonness can fail for affine PI algebras: Example 4.12 is a formally real affine PI R \mathbb{R} R k k k k k k R R R ∗ * ∗ R → ( M n ( K ) , T ) R \to(M_n(K),\mathsf{T}) R → ( M n ( K ) , T ) K K K k k k
Part (1) is proved by central localization (Theorem 4.3). Parts (2) and (3) follow from Theorem 4.7, which applies to rings whose finitely generated prime modules embed in products of their annihilator quotients; PI rings have this property (Lemma 4.8). Parts (4) and (5) are Corollaries 4.9 and 4.21, respectively.
Free algebras
For free associative algebras, we must restrict ourselves to finitely generated submodules of finite free modules. We obtain a version of Hilbert’s Nullstellensatz:
Theorem D. Let F = k ⟨ x 1 , … , x g ⟩ F = k\langle x_1,\ldots,x_g\rangle F = k ⟨ x 1 , … , x g ⟩ k k k g < ∞ g < \infty g < ∞
Every finitely presented semiprime left F F F N ⊆ F n N \subseteq F^n N ⊆ F n
A finitely generated submodule N ⊆ F n N \subseteq F^n N ⊆ F n F F F F n F^n F n k k k
If N N N F n F^n F n rad N \operatorname{rad}N rad N rad N / N \operatorname{rad}N/N rad N / N rad N \operatorname{rad}N rad N k = Q k = \mathbb{Q} k = Q k k k
The finitely generated maximal submodules in part (2) can be described explicitly. Choose a finite-dimensional irreducible representation π : F → End k ( V ) \pi:F\to\operatorname{End}_k(V) π : F → End k ( V ) v 1 , … , v n ∈ V v_1,\ldots,v_n\in V v 1 , … , v n ∈ V F F F
ε π , v : F n ⟶ V , ( f 1 , … , f n ) ⟼ ∑ j π ( f j ) v j . (1) \varepsilon_{\pi,v}:F^n\longrightarrow V,\qquad(f_1,\ldots,f_n)\longmapsto\sum_j\pi(f_j)v_j. \tag*{(1)} ε π , v : F n ⟶ V , ( f 1 , … , f n ) ⟼ j ∑ π ( f j ) v j . (1) The kernels L π , v = ker ε π , v L_{\pi,v}=\ker\varepsilon_{\pi,v} L π , v = ker ε π , v k k k Proposition 5.5 ).
It is classical that every finitely generated left ideal in a free associative algebra is an intersection of left ideals of finite codimension: see Rosenmann–Rosset [33 Theorem 4.3] (they work with right ideals), and Bergman’s Nullstellensatz in [20 ] . Those statements allow representations that need not be irreducible. Restricting to irreducible representations gives the semiprime hull instead of the original ideal, a question raised in [10 Section 2.3] . Theorem D answers this question and treats submodules of F n F^n F n n = 1 n=1 n = 1
Free algebras with involution
Parts (2) and (3) of Theorem E follow from [8 , 19 , 30 ] , as outlined in Remark 6.11. Our contribution here is that we extend part (1) to arbitrary fields and involutions and that we reprove part (2) and the finite-generation and degree-bound assertions of part (3) by algebraic methods, without using the GNS construction or Gröbner bases.
Theorem E. Let k k k F = k ⟨ x 1 , … , x g ⟩ F=k\langle x_1,\ldots,x_g\rangle F = k ⟨ x 1 , … , x g ⟩ g < ∞ g<\infty g < ∞ k k k
Every finitely presented real left F F F F n F^n F n
For (2) and (3), suppose that k = R k=\mathbb{R} k = R k = C k=\mathbb{C} k = C ∗ * ∗ x 1 , … , x g x_1,\ldots,x_g x 1 , … , x g
[8 , 19 , 30 ] A finitely generated submodule N ⊆ F n N\subseteq F^n N ⊆ F n F F F F n F^n F n k k k π \pi π ∗ * ∗
[8 , 19 , 30 ] The real radical r e N {}^{\mathrm{re}}\!\sqrt{N} re N N ⊆ F n N\subseteq F^n N ⊆ F n N N N d ≥ 1 d\geq1 d ≥ 1 n ∑ j = 0 d − 1 g j n\sum_{j=0}^{d-1}g^j n ∑ j = 0 d − 1 g j d d d
Related work and organization
Applications of this theory to polynomial and group rings will be developed in separate work.
Sections 2 and 3 prove Theorems A and B, respectively. Section 4 proves Theorem C. Sections 5 and 6 prove Theorems D and E, except for the algorithmic assertions. Appendix A gives the prime-radical algorithm and recalls the established real-radical algorithm.
Prime separation
Theorem A gives two conditions on the ring under which every semiprime submodule is an intersection of prime submodules. We also prove that Noetherian modules over arbitrary rings are prime-separated.
Prime and semiprime submodules
Prime and semiprime modules extend the corresponding notions for left ideals. The definitions below let us treat all modules in a single statement and recover the left-ideal result as a special case.
Definition 2.1 (Prime and semiprime modules). For a left R R R L L L annihilator is
Ann R ( L ) = { a ∈ R : a ℓ = 0 for every ℓ ∈ L } . \operatorname{Ann}_{R}(L)=\{a\in R:a\ell=0\text{ for every }\ell\in L\}. Ann R ( L ) = { a ∈ R : a ℓ = 0 for every ℓ ∈ L } . It is a two-sided ideal of R R R R R R Q Q Q prime if every nonzero submodule L ⊆ Q L\subseteq Q L ⊆ Q Ann R ( L ) = Ann R ( Q ) \operatorname{Ann}_{R}(L)=\operatorname{Ann}_{R}(Q) Ann R ( L ) = Ann R ( Q ) R R R Q Q Q semiprime if
L ∩ Ann R ( L ) Q = 0 for every submodule L ⊆ Q . L\cap\operatorname{Ann}_{R}(L)Q=0\qquad\text{for every submodule }L\subseteq Q. L ∩ Ann R ( L ) Q = 0 for every submodule L ⊆ Q . A submodule N ⊆ M N\subseteq M N ⊆ M prime or semiprime if M / N M/N M / N
Lemma 2.2 gives practical tests for these definitions and checks that they recover the corresponding notions for left ideals.
Lemma 2.2 (Characterizations). Let Q , M Q,M Q , M R R R N ⊆ M N\subseteq M N ⊆ M
(1) A nonzero Q Q Q
a R q = 0 ⟹ q = 0 or a Q = 0 ( a ∈ R , q ∈ Q ) . aRq=0\quad\Longrightarrow\quad q=0\text{ or }aQ=0\qquad(a\in R,\ q\in Q). a R q = 0 ⟹ q = 0 or a Q = 0 ( a ∈ R , q ∈ Q ) . (2) The module Q Q Q
q ∈ Ann R ( R q ) Q ⟹ q = 0 ( q ∈ Q ) , q\in\operatorname{Ann}_{R}(Rq)Q\quad\Longrightarrow\quad q=0\qquad(q\in Q), q ∈ Ann R ( R q ) Q ⟹ q = 0 ( q ∈ Q ) , or, equivalently, L ⊆ Ann R ( L ) Q L\subseteq\operatorname{Ann}_{R}(L)Q L ⊆ Ann R ( L ) Q L = 0 L=0 L = 0 L ⊆ Q L\subseteq Q L ⊆ Q
In particular:
(3) The submodule N N N N ≠ M N\ne M N = M
a R m ⊆ N ⟹ m ∈ N or a M ⊆ N ( a ∈ R , m ∈ M ) . aRm\subseteq N\quad\Longrightarrow\quad m\in N\text{ or }aM\subseteq N\qquad(a\in R,\ m\in M). a R m ⊆ N ⟹ m ∈ N or a M ⊆ N ( a ∈ R , m ∈ M ) . It is semiprime if and only if, for every finite sum m = ∑ i a i m i m=\sum_i a_i m_i m = ∑ i a i m i a i ∈ R a_i\in R a i ∈ R m i ∈ M m_i\in M m i ∈ M
a i R m ⊆ N for all i ⟹ m ∈ N . a_iRm\subseteq N\text{ for all }i\quad\Longrightarrow\quad m\in N. a i R m ⊆ N for all i ⟹ m ∈ N . (4) A left ideal is prime (respectively semiprime) as a submodule of R R {}_R R R R
Proof. (1) The criterion says that every nonzero cyclic submodule has annihilator Ann R ( Q ) \operatorname{Ann}_{R}(Q) Ann R ( Q )
(2) Semiprimeness gives both implications. Conversely, the condition on submodules, applied to H = L ∩ Ann R ( L ) Q H=L\cap\operatorname{Ann}_{R}(L)Q H = L ∩ Ann R ( L ) Q H = 0 H=0 H = 0 H ⊆ Ann R ( L ) Q ⊆ Ann R ( H ) Q H\subseteq\operatorname{Ann}_{R}(L)Q\subseteq\operatorname{Ann}_{R}(H)Q H ⊆ Ann R ( L ) Q ⊆ Ann R ( H ) Q H = 0 H=0 H = 0 q ∈ H q\in H q ∈ H Ann R ( R q ) Q \operatorname{Ann}_{R}(Rq)Q Ann R ( R q ) Q Ann R ( L ) ⊆ Ann R ( R q ) \operatorname{Ann}_{R}(L)\subseteq\operatorname{Ann}_{R}(Rq) Ann R ( L ) ⊆ Ann R ( R q )
(3) Apply (1) and (2) to M / N M/N M / N q ∈ Ann R ( R q ) ( M / N ) q\in\operatorname{Ann}_{R}(Rq)(M/N) q ∈ Ann R ( R q ) ( M / N ) q = ∑ i a i ( m i + N ) q=\sum_i a_i(m_i+N) q = ∑ i a i ( m i + N ) a i R q = 0 a_iRq=0 a i R q = 0 m = ∑ i a i m i m=\sum_i a_im_i m = ∑ i a i m i
(4) The prime criterion in (3) implies the left-ideal criterion. Conversely, if a R b ⊆ I aRb\subseteq I a R b ⊆ I b ∉ I b\notin I b ∈ / I a c , b ac,b a c , b c ∈ R c\in R c ∈ R a R ⊆ I aR\subseteq I a R ⊆ I m = a ⋅ 1 m=a\cdot1 m = a ⋅ 1 m = ∑ i a i b i m=\sum_i a_i b_i m = ∑ i a i b i a i R m ⊆ I a_i Rm\subseteq I a i R m ⊆ I m R m ⊆ ∑ i a i R m ⊆ I mRm\subseteq\sum_i a_i Rm\subseteq I m R m ⊆ ∑ i a i R m ⊆ I m ∈ I m\in I m ∈ I □ \square □
For modules, the term semiprime has several meanings in the literature. We use the same notion as in [7 ] (Lemma 2.2(2)); Jiráško calls it pseudo-semiprimeness [22 ] (Proposition 1.2(vi)). See also the discussion preceding [26 ] (Proposition 5.5). By Lemma 2.2(2), our notion implies semiprimeness in the sense of Dauns [12 ] , namely a R a m = 0 ⇒ a m = 0 aRam=0\Rightarrow am=0 a R am = 0 ⇒ am = 0 [26 p. 93] .
The following closure properties let us build semiprime modules from simpler ones.
Lemma 2.3 (Prime and semiprime modules). Let R R R
(1) Every simple R R R R R R
(2) Every nonzero submodule of a prime module is a prime module.
(3) Every submodule of a semiprime module is a semiprime module.
(4) Every subdirect product of semiprime modules is semiprime. In particular, products and direct sums of semiprime modules, and hence of prime modules, are semiprime.
Proof. (1) A simple module has no nonzero submodule other than itself, so it is prime. If Q Q Q 0 ≠ L ⊆ Q 0\neq L\subseteq Q 0 = L ⊆ Q Ann R ( L ) Q = Ann R ( Q ) Q = 0 \operatorname{Ann}_R(L)Q=\operatorname{Ann}_R(Q)Q=0 Ann R ( L ) Q = Ann R ( Q ) Q = 0 L = 0 L=0 L = 0 Q Q Q
(2) If 0 ≠ L ⊆ Q 0\neq L\subseteq Q 0 = L ⊆ Q Q Q Q H ⊆ L H\subseteq L H ⊆ L Ann R ( H ) = Ann R ( Q ) = Ann R ( L ) \operatorname{Ann}_R(H)=\operatorname{Ann}_R(Q)=\operatorname{Ann}_R(L) Ann R ( H ) = Ann R ( Q ) = Ann R ( L ) L L L
(3) If N ⊆ Q N\subseteq Q N ⊆ Q Q Q Q L ⊆ N L\subseteq N L ⊆ N
L ∩ Ann R ( L ) N ⊆ L ∩ Ann R ( L ) Q = 0 L\cap\operatorname{Ann}_R(L)N\subseteq L\cap\operatorname{Ann}_R(L)Q=0 L ∩ Ann R ( L ) N ⊆ L ∩ Ann R ( L ) Q = 0 so N N N
(4) Let V V V V λ V_\lambda V λ f λ : V ↠ V λ f_\lambda:V\twoheadrightarrow V_\lambda f λ : V ↠ V λ L ⊆ V L\subseteq V L ⊆ V Ann R ( L ) ⊆ Ann R ( f λ ( L ) ) \operatorname{Ann}_R(L)\subseteq\operatorname{Ann}_R(f_\lambda(L)) Ann R ( L ) ⊆ Ann R ( f λ ( L ))
f λ ( L ∩ Ann R ( L ) V ) ⊆ f λ ( L ) ∩ Ann R ( f λ ( L ) ) V λ = 0. f_\lambda\left(L\cap\operatorname{Ann}_R(L)V\right)\subseteq f_\lambda(L)\cap\operatorname{Ann}_R(f_\lambda(L))V_\lambda=0. f λ ( L ∩ Ann R ( L ) V ) ⊆ f λ ( L ) ∩ Ann R ( f λ ( L )) V λ = 0. Since the maps jointly detect zero, L ∩ Ann R ( L ) V = 0 L\cap\operatorname{Ann}_R(L)V=0 L ∩ Ann R ( L ) V = 0 □ \square □
Primeness and semiprimeness of submodules are defined by the corresponding properties of their factor modules. Therefore Lemma 2.3 immediately implies Lemma 2.4 .
Lemma 2.4 (Prime and semiprime submodules). Let M M M R R R
(1) Every maximal submodule of M M M M M M
(2) Every intersection of semiprime submodules of M M M
(3) If N ⊆ L ⊆ M N\subseteq L\subseteq M N ⊆ L ⊆ M L L L M M M L / N L/N L / N M / N M/N M / N
Proof. For (1), apply Lemma 2.3(1) to quotient modules. For (2), M / ⋂ λ N λ M/\bigcap_\lambda N_\lambda M / ⋂ λ N λ M / N λ M/N_\lambda M / N λ ( M / N ) / ( L / N ) ≅ M / L (M/N)/(L/N)\cong M/L ( M / N ) / ( L / N ) ≅ M / L □ \square □
Annihilators connect prime and semiprime modules with the familiar two-sided ideal notions.
Lemma 2.5 (Prime and semiprime annihilators).
(1) The annihilator of a prime module is a prime two-sided ideal.
(2) The annihilator of a semiprime module is a semiprime two-sided ideal. In particular, the annihilator of every submodule of a semiprime module is semiprime.
Proof. (1) Let Q Q Q T = Ann R ( Q ) T=\operatorname{Ann}_{R}(Q) T = Ann R ( Q ) Q ≠ 0 Q\ne0 Q = 0 A , B A,B A , B A B ⊆ T AB\subseteq T A B ⊆ T B ⊈ T B\not\subseteq T B ⊆ T B Q ≠ 0 BQ\ne0 B Q = 0 Ann R ( B Q ) = T \operatorname{Ann}_{R}(BQ)=T Ann R ( B Q ) = T A ( B Q ) = 0 A(BQ)=0 A ( B Q ) = 0 A ⊆ T A\subseteq T A ⊆ T T T T
(2) Let Q Q Q A A A A 2 Q = 0 A^{2}Q=0 A 2 Q = 0 H = A Q H=AQ H = A Q A H = 0 AH=0 A H = 0 H = A Q ⊆ Ann R ( H ) Q H=AQ\subseteq\operatorname{Ann}_{R}(H)Q H = A Q ⊆ Ann R ( H ) Q H = 0 H=0 H = 0 A ⊆ Ann R ( Q ) A\subseteq\operatorname{Ann}_{R}(Q) A ⊆ Ann R ( Q ) Ann R ( Q ) \operatorname{Ann}_{R}(Q) Ann R ( Q )
The separation argument will use the following way of extending a prime module.
Lemma 2.6 (Essential extensions of prime modules). Let U U U essential submodule of M M M U ∩ H ≠ 0 U\cap H\ne0 U ∩ H = 0 H ⊆ M H\subseteq M H ⊆ M U U U Ann R ( M ) = Ann R ( U ) \operatorname{Ann}_{R}(M)=\operatorname{Ann}_{R}(U) Ann R ( M ) = Ann R ( U ) M M M
Proof. For 0 ≠ H ⊆ M 0\ne H\subseteq M 0 = H ⊆ M H ∩ U ≠ 0 H\cap U\ne0 H ∩ U = 0 U U U
Ann R ( M ) ⊆ Ann R ( H ) ⊆ Ann R ( H ∩ U ) = Ann R ( U ) = Ann R ( M ) . \operatorname{Ann}_{R}(M)\subseteq\operatorname{Ann}_{R}(H)\subseteq\operatorname{Ann}_{R}(H\cap U)=\operatorname{Ann}_{R}(U)=\operatorname{Ann}_{R}(M). Ann R ( M ) ⊆ Ann R ( H ) ⊆ Ann R ( H ∩ U ) = Ann R ( U ) = Ann R ( M ) . Thus every nonzero submodule of M M M M M M M M M U U U Ann R ( M ) ⊆ Ann R ( U ) = 0 \operatorname{Ann}_{R}(M)\subseteq\operatorname{Ann}_{R}(U)=0 Ann R ( M ) ⊆ Ann R ( U ) = 0
Prime separation
We now formulate prime separation for modules. The observations and the Separation Lemma below apply over any ring.
Definition 2.7 (Prime separation). A left R R R M M M prime-separated if every semiprime submodule is an intersection of prime submodules. Equivalently, for every semiprime N ⊆ M N\subseteq M N ⊆ M m ∉ N m\notin N m ∈ / N N N N m m m prime-separated if its regular left module is. We state explicitly when the property holds for all left modules.
When every R R R M = R R M={}_{R}R M = R R J J J semiprime hull , is then the intersection of all prime left ideals containing J J J
Write β ( R ) \beta(R) β ( R ) lower nilradical (or two-sided prime radical) of R R R P P P Ann R ( R / P ) \operatorname{Ann}_{R}(R/P) Ann R ( R / P )
Part (3) of the next lemma uses Theorem A, proved below independently of this lemma.
Lemma 2.8 (Basic constructions).
Prime-separated rings are closed under quotients and finite products. If I ⊆ β ( R ) I\subseteq\beta(R) I ⊆ β ( R ) R R R R / I R/I R / I
For any rings A , B A,B A , B ( A , B ) (A,B) ( A , B ) M M M T = ( A M 0 B ) T=\begin{pmatrix}A&M\\0&B\end{pmatrix} T = ( A 0 M B ) A A A B B B
If R / β ( R ) R/\beta(R) R / β ( R ) R R R
Proof. (1) Restriction along a quotient map preserves primeness and semiprimeness, giving closure under quotients. If I ⊆ β ( R ) I\subseteq\beta(R) I ⊆ β ( R ) L ⊆ V L\subseteq V L ⊆ V I V ⊆ L IV\subseteq L I V ⊆ L Ann R ( V / L ) \operatorname{Ann}_{R}(V/L) Ann R ( V / L ) V → V / I V V\to V/IV V → V / I V T T T A × B A \times B A × B T T T A A A B B B
(3) The first assertion follows from Theorem A and (1). To prove closure, first suppose that R R R S = M n ( R ) S=M_n(R) S = M n ( R ) b = ( b i j ) ∈ S b=(b_{ij})\in S b = ( b ij ) ∈ S S b S = M n ( I ) SbS=M_n(I) S b S = M n ( I ) I = ∑ i , j R b i j R I=\sum_{i,j} Rb_{ij}R I = ∑ i , j R b ij R R R R I = ∑ ℓ = 1 d R t ℓ I=\sum_{\ell=1}^{d}Rt_{\ell} I = ∑ ℓ = 1 d R t ℓ
M n ( I ) = ∑ ℓ = 1 d ∑ j = 1 n S ( t ℓ E 1 j ) . M_n(I)=\sum_{\ell=1}^{d}\sum_{j=1}^{n}S(t_{\ell}E_{1j}). M n ( I ) = ℓ = 1 ∑ d j = 1 ∑ n S ( t ℓ E 1 j ) . so matrix rings also preserve (ii). Apply these observations to R / β ( R ) R/\beta(R) R / β ( R ) β ( M n ( R ) ) = M n ( β ( R ) ) \beta(M_n(R))=M_n(\beta(R)) β ( M n ( R )) = M n ( β ( R )) T / β ( T ) ≅ A / β ( A ) × B / β ( B ) T/\beta(T)\cong A/\beta(A)\times B/\beta(B) T / β ( T ) ≅ A / β ( A ) × B / β ( B ) R R R R / β ( R ) R/\beta(R) R / β ( R )
Thus Question 1 reduces to semiprime rings.
Lemma 2.9 (Separation). Let N N N semiprime submodule of M M M x ∈ M ∖ N x\in M\setminus N x ∈ M ∖ N ( N + R x ) / N (N+Rx)/N ( N + R x ) / N nonzero prime submodule . Then some prime submodule of M M M N N N x x x
Proof. Choose N ⊊ L ⊆ N + R x N\subsetneq L\subseteq N+Rx N ⊊ L ⊆ N + R x L / N L/N L / N T = Ann R ( L / N ) T=\operatorname{Ann}_{R}(L/N) T = Ann R ( L / N ) M / N M/N M / N ( L / N ) ∩ T ( M / N ) = 0 (L/N)\cap T(M/N)=0 ( L / N ) ∩ T ( M / N ) = 0 L ∩ ( N + T M ) = N L\cap(N+TM)=N L ∩ ( N + T M ) = N
Choose P P P N + T M N+TM N + T M P ∩ L = N P\cap L=N P ∩ L = N P P P N + T M N+TM N + T M x ∈ P x\in P x ∈ P L ⊆ N + R x ⊆ P L\subseteq N+Rx\subseteq P L ⊆ N + R x ⊆ P P ∩ L = N P\cap L=N P ∩ L = N L ≠ N L\ne N L = N x ∉ P x\notin P x ∈ / P
The image L ‾ = ( L + P ) / P ≅ L / N \overline{L}=(L+P)/P\cong L/N L = ( L + P ) / P ≅ L / N M / P M/P M / P P ⊊ U ⊆ M P\subsetneq U\subseteq M P ⊊ U ⊆ M U ∩ L ⊋ N U\cap L\supsetneq N U ∩ L ⊋ N U / P U/P U / P L ‾ \overline{L} L L ‾ \overline{L} L T T T T M ⊆ P TM\subseteq P T M ⊆ P
T ⊆ Ann R ( M / P ) ⊆ Ann R ( L ‾ ) = T . T\subseteq\operatorname{Ann}_{R}(M/P)\subseteq\operatorname{Ann}_{R}(\overline{L})=T. T ⊆ Ann R ( M / P ) ⊆ Ann R ( L ) = T . Lemma 2.6 makes M / P M/P M / P
ACC on two-sided semiprime ideals
An ascending chain condition on semiprime two-sided ideals is enough to ensure prime separation for every module. This covers, in particular, left Noetherian rings and affine PI algebras. We also show that locally Noetherian modules are prime-separated over any ring.
Lemma 2.10 (Existence of prime submodules). Let Q Q Q semiprime and 0 ≠ q ∈ Q 0\ne q\in Q 0 = q ∈ Q R q Rq R q nonzero prime submodule in each of the following cases:
R R R
R q Rq R q nonzero uniform submodule , in particular if R q Rq R q
Proof. (1) Each Ann R ( L ) \operatorname{Ann}_{R}(L) Ann R ( L ) 0 ≠ L ⊆ R q 0\ne L\subseteq Rq 0 = L ⊆ R q Lemma 2.5 (2). Choose L L L Ann R ( L ) \operatorname{Ann}_{R}(L) Ann R ( L ) 0 ≠ H ⊆ L 0\ne H\subseteq L 0 = H ⊆ L Ann R ( H ) = Ann R ( L ) \operatorname{Ann}_{R}(H)=\operatorname{Ann}_{R}(L) Ann R ( H ) = Ann R ( L ) L L L
(2) Let 0 ≠ U ⊆ R q 0\ne U\subseteq Rq 0 = U ⊆ R q R q Rq R q 0 ≠ H ⊆ U 0\ne H\subseteq U 0 = H ⊆ U
H ∩ Ann R ( H ) U ⊆ H ∩ Ann R ( H ) Q = 0. H\cap\operatorname{Ann}_{R}(H)U\subseteq H\cap\operatorname{Ann}_{R}(H)Q=0. H ∩ Ann R ( H ) U ⊆ H ∩ Ann R ( H ) Q = 0. Since H ≠ 0 H\ne0 H = 0 H H H Ann R ( H ) U \operatorname{Ann}_{R}(H)U Ann R ( H ) U U U U Ann R ( H ) U = 0 \operatorname{Ann}_{R}(H)U=0 Ann R ( H ) U = 0 Ann R ( H ) ⊆ Ann R ( U ) \operatorname{Ann}_{R}(H)\subseteq\operatorname{Ann}_{R}(U) Ann R ( H ) ⊆ Ann R ( U ) H ⊆ U H\subseteq U H ⊆ U U U U U U U U U U
Proof of Theorem A under hypothesis (i). Let N ⊆ M N \subseteq M N ⊆ M x ∉ N x \notin N x ∈ / N Q = M / N Q=M/N Q = M / N q = x + N q=x+N q = x + N N N N x x x □ \square □
Corollary 2.11 (Locally Noetherian modules). Over an arbitrary ring, every locally Noetherian module, that is, every module whose cyclic submodules are Noetherian, is prime-separated. This applies in particular to Noetherian modules and to finite-dimensional modules over any algebra over a field.
Proof. Let N N N M M M x ∉ N x \notin N x ∈ / N ( N + R x ) / N (N+Rx)/N ( N + R x ) / N R x Rx R x M / N M/N M / N □ \square □
Corollary 2.12 (Classes of rings). Each of the following rings satisfies ACC on semiprime two-sided ideals and hence the conclusions of Theorem A for every left module:
a left Noetherian ring;
an affine PI algebra over a field;
a ring with left Krull dimension;
a commutative ring whose spectrum is Noetherian.
Proof. (1) Every ascending chain of two-sided ideals is an ascending chain of left ideals, so it stabilizes.
(2) Affine PI algebras satisfy ACC on semiprime two-sided ideals by [2 Corollary 1, p. 182] .
(3) Apply [16 Theorem 7.6] to R op R^{\operatorname{op}} R op R R R R R R
(4) In a commutative ring, semiprime ideals are radical ideals. The correspondence I ↦ V ( I ) I \mapsto V(I) I ↦ V ( I ) Spec R \operatorname{Spec} R Spec R
Theorem A now applies in each case. □ \square □
Theorem 4.3 will extend prime separation to all PI rings.
Principal two-sided ideals as left ideals
The role of hypothesis (ii) of Theorem A is to ensure that unions of nonempty chains of semiprime submodules are semiprime. We first show that this chain-union property implies prime separation, and then verify it under (ii). The hypothesis covers every commutative ring and tensor products over a commutative ring k k k k k k k k k
Lemma 2.13 (Annihilator quotients). Let Q Q Q I I I Q / ( 0 : Q I ) Q/(0 :_Q I) Q / ( 0 : Q I ) ( 0 : Q I ) = { q ∈ Q : I q = 0 } (0 :_Q I)=\{q \in Q : Iq=0\} ( 0 : Q I ) = { q ∈ Q : I q = 0 }
Proof. Put K = ( 0 : Q I ) K=(0 :_Q I) K = ( 0 : Q I ) L ⊇ K L \supseteq K L ⊇ K J = Ann R ( L / K ) J=\operatorname{Ann}_R(L/K) J = Ann R ( L / K ) J L ⊆ K JL \subseteq K J L ⊆ K I J ⊆ Ann R ( L ) IJ \subseteq\operatorname{Ann}_R(L) I J ⊆ Ann R ( L ) ( I J ) 2 ⊆ J ( I J ) (IJ)^2 \subseteq J(IJ) ( I J ) 2 ⊆ J ( I J ) J I ⊆ Ann R ( I L ) JI \subseteq\operatorname{Ann}_R(I L) J I ⊆ Ann R ( I L ) J I L = 0 JIL=0 J I L = 0 J ⊆ Ann R ( I L ) J \subseteq\operatorname{Ann}_R(IL) J ⊆ Ann R ( I L ) x ∈ L ∩ ( J Q + K ) x \in L \cap(JQ+K) x ∈ L ∩ ( J Q + K ) x = y + k x=y+k x = y + k y ∈ J Q y \in JQ y ∈ J Q k ∈ K k \in K k ∈ K I x = I y ⊆ I J Q ⊆ J Q Ix=Iy \subseteq IJQ \subseteq JQ I x = I y ⊆ I J Q ⊆ J Q I x ⊆ I L Ix \subseteq IL I x ⊆ I L
I x ⊆ I L ∩ J Q ⊆ I L ∩ Ann R ( I L ) Q = 0. Ix \subseteq IL \cap JQ \subseteq IL \cap\operatorname{Ann}_R(IL)Q=0. I x ⊆ I L ∩ J Q ⊆ I L ∩ Ann R ( I L ) Q = 0. Thus L ∩ ( J Q + K ) ⊆ K L \cap(JQ+K) \subseteq K L ∩ ( J Q + K ) ⊆ K J ( Q / K ) = ( J Q + K ) / K J(Q/K)=(JQ+K)/K J ( Q / K ) = ( J Q + K ) / K Q / K Q/K Q / K □ \square □
The preceding quotient construction lets us show that a semiprime submodule maximal subject to avoiding an element is prime.
Lemma 2.14 (Maximal semiprime submodules). Let x ∈ M x \in M x ∈ M N N N M M M x x x N N N M M M M M M
Proof. Put Q = M / N Q=M/N Q = M / N q = x + N q=x+N q = x + N 0 ≠ H ⊆ R q 0 \ne H \subseteq Rq 0 = H ⊆ R q J = Ann R ( H ) J=\operatorname{Ann}_{R}(H) J = Ann R ( H ) ( 0 : Q J ) (0 :_{Q} J) ( 0 : Q J ) H ≠ 0 H \ne0 H = 0 Q Q Q M M M N N N x x x J q = 0 Jq=0 J q = 0 J J J J R q ⊆ J q = 0 JRq \subseteq Jq=0 J R q ⊆ J q = 0 Ann R ( H ) ⊆ Ann R ( R q ) \operatorname{Ann}_{R}(H) \subseteq\operatorname{Ann}_{R}(Rq) Ann R ( H ) ⊆ Ann R ( R q ) R q Rq R q N N N x x x N N N
Under the chain-union hypothesis, Zorn’s lemma gives such an N N N
Proof of Theorem A under hypothesis (ii). By Lemma 2.14 , it suffices to show that a nonempty chain ( N λ ) (N_{\lambda}) ( N λ ) N N N
x − ∑ i = 1 r a i m i ∈ N , a i R x ⊆ N ( 1 ≤ i ≤ r ) . x-\sum_{i=1}^{r}a_{i}m_{i}\in N,\qquad a_{i}Rx\subseteq N\quad(1\leq i\leq r). x − i = 1 ∑ r a i m i ∈ N , a i R x ⊆ N ( 1 ≤ i ≤ r ) . Since the elements a i s a_{i}s a i s s ∈ R s\in R s ∈ R R a i R Ra_{i}R R a i R R a i R = ∑ j = 1 d R a i s i j Ra_{i}R=\sum_{j=1}^{d}Ra_{i}s_{ij} R a i R = ∑ j = 1 d R a i s ij x − ∑ i a i m i x-\sum_{i}a_{i}m_{i} x − ∑ i a i m i a i s i j x a_{i}s_{ij}x a i s ij x N λ N_{\lambda} N λ R a i R x = ∑ j R a i s i j x ⊆ N λ Ra_{i}Rx=\sum_{j}Ra_{i}s_{ij}x\subseteq N_{\lambda} R a i R x = ∑ j R a i s ij x ⊆ N λ a i R x ⊆ N λ a_{i}Rx\subseteq N_{\lambda} a i R x ⊆ N λ M / N λ M/N_{\lambda} M / N λ x x x Ann R ( R ( x + N λ ) ) ( M / N λ ) \operatorname{Ann}_{R}(R(x+N_{\lambda}))(M/N_{\lambda}) Ann R ( R ( x + N λ )) ( M / N λ ) x ∈ N λ ⊆ N x\in N_{\lambda}\subseteq N x ∈ N λ ⊆ N N N N
Corollary 2.15 (Classes of rings). Each of the following rings R R R
a commutative ring;
a ring finitely generated as a module over its centre (e.g. an Azumaya algebra);
a ring generated by a left Noetherian subring and a central subring (e.g. B ⊗ k S B\otimes_{k}S B ⊗ k S k k k B B B k k k S S S k k k
a biregular ring (Example 2.17).
Consequently, every left module over these rings is prime-separated.
Proof. For (1), R a R = R a RaR=Ra R a R = R a b 1 , … , b d b_{1},\ldots,b_{d} b 1 , … , b d R R R Z ( R ) Z(R) Z ( R ) R a R = ∑ j = 1 d R a b j RaR=\sum_{j=1}^{d}Rab_{j} R a R = ∑ j = 1 d R a b j B B B C C C R = B C R=BC R = B C b c bc b c a = ∑ i = 1 d b i c i a=\sum_{i=1}^{d}b_{i}c_{i} a = ∑ i = 1 d b i c i B a B ⊆ ∑ i B c i BaB\subseteq\sum_{i}Bc_{i} B a B ⊆ ∑ i B c i B B B B a B = ∑ j = 1 e B u j BaB=\sum_{j=1}^{e}Bu_{j} B a B = ∑ j = 1 e B u j u j u_{j} u j R a R = C ( B a B ) = ∑ j = 1 e R u j RaR=C(BaB)=\sum_{j=1}^{e}Ru_{j} R a R = C ( B a B ) = ∑ j = 1 e R u j B ⊗ 1 B\otimes1 B ⊗ 1 1 ⊗ S 1\otimes S 1 ⊗ S B B B R a R = R e RaR=Re R a R = R e e e e
The two hypotheses are independent: k [ t 1 , t 2 , … ] k[t_{1},t_{2},\ldots] k [ t 1 , t 2 , … ] Remark 4.10 satisfies (i) but not (ii).
Example 2.16 (FC-group crossed products). Let B B B G G G T = B ∗ G T=B*G T = B ∗ G T = ⨁ g ∈ G B u g T=\bigoplus_{g\in G}Bu_{g} T = ⨁ g ∈ G B u g u g u_{g} u g a ∈ T a\in T a ∈ T F F F F G = ⋃ g ∈ G g F g − 1 F^{G}=\bigcup_{g\in G}gFg^{-1} F G = ⋃ g ∈ G g F g − 1 B B B
V = ∑ g ∈ G B ( u g − 1 a u g ) B ⊆ ⨁ h ∈ F G B u h V=\sum_{g\in G}B(u_{g}^{-1}au_{g})B\subseteq\bigoplus_{h\in F^{G}}Bu_{h} V = g ∈ G ∑ B ( u g − 1 a u g ) B ⊆ h ∈ F G ⨁ B u h is finitely generated on the left. Since T a T = T V TaT=TV T a T = T V u g u_g u g a b u g = u g ( u g − 1 a u g ) ( u g − 1 b u g ) abu_g=u_g(u_g^{-1}au_g)(u_g^{-1}bu_g) ab u g = u g ( u g − 1 a u g ) ( u g − 1 b u g ) b ∈ B b\in B b ∈ B B B B
Example 2.17 (Biregular rings). A ring is biregular if every principal two-sided ideal is generated by a central idempotent. Let k k k
A = k 1 + ⨁ n ≥ 2 M n ( k ) ⊂ ∏ n ≥ 2 M n ( k ) A=k1+\bigoplus_{n\geq2}M_n(k)\subset\prod_{n\geq2}M_n(k) A = k 1 + n ≥ 2 ⨁ M n ( k ) ⊂ n ≥ 2 ∏ M n ( k ) every a a a A a A = A e a AaA=Ae_a A a A = A e a e a e_a e a A A A M n ( k ) M_n(k) M n ( k )
Now put U = ( k A 0 A ) U=\begin{pmatrix} k & A \\ 0 & A \end{pmatrix} U = ( k 0 A A ) U / β ( U ) ≅ k × A U/\beta(U)\cong k\times A U / β ( U ) ≅ k × A U U U Lemma 2.8 . Yet U U U A A A U e 11 U ≅ k ⊕ A Ue_{11}U\cong k\oplus A U e 11 U ≅ k ⊕ A k k k U U U
Rings with involution
The first three subsections prove the three parts of Theorem B: real two-sided prime separation, the Jordan-plane counterexample, and real prime separation over hereditary Noetherian prime rings. The last subsection studies semireality and the existence of real maximal left ideals. Section 4 then develops the PI case.
Real two-sided ideals
Let R R R a ↦ a ∗ a\mapsto a^* a ↦ a ∗ I I I ∑ i a i ∗ a i ∈ I + I ∗ \sum_i a_i^*a_i\in I+I^* ∑ i a i ∗ a i ∈ I + I ∗ a i ∈ I a_i\in I a i ∈ I weakly real if ∑ i a i ∗ a i ∈ I \sum_i a_i^*a_i\in I ∑ i a i ∗ a i ∈ I a i ∈ I a_i\in I a i ∈ I a i ∈ R a_i\in R a i ∈ R R R R I ⊆ I + I ∗ I\subseteq I+I^* I ⊆ I + I ∗ I I I a ∈ I a\in I a ∈ I ( a ∗ ) ∗ a ∗ = a a ∗ ∈ I (a^*)^*a^*=aa^*\in I ( a ∗ ) ∗ a ∗ = a a ∗ ∈ I a ∗ ∈ I a^*\in I a ∗ ∈ I I = I ∗ I=I^* I = I ∗ I + I ∗ = I I+I^*=I I + I ∗ = I Proposition 3.12 ), but not in general ((Examples 3.3 and 6.6)). Weak reality is the notion called “real” in [4 §1.3] ; whether the two definitions agree for all left ideals was asked in [4 Remark 5] .
In a real ∗ * ∗ T = Ann R ( J ) T=\operatorname{Ann}_R(J) T = Ann R ( J ) J J J ∑ i a i ∗ a i ∈ T \sum_i a_i^*a_i\in T ∑ i a i ∗ a i ∈ T j ∈ J j\in J j ∈ J
∑ i ( a i j ) ∗ ( a i j ) = j ∗ ( ∑ i a i ∗ a i ) j = 0. \sum_i(a_i j)^*(a_i j)=j^*\left(\sum_i a_i^*a_i\right)j=0. i ∑ ( a i j ) ∗ ( a i j ) = j ∗ ( i ∑ a i ∗ a i ) j = 0. Reality of R R R a i J = 0 a_iJ=0 a i J = 0 a i ∈ T a_i\in T a i ∈ T T T T
Theorem 3.1 (Real two-sided ideals). In every ∗ * ∗
Proof. For two-sided ideals, reality and weak reality agree; hence a union of a chain of real two-sided ideals is real. Let I I I a ∉ I a\notin I a ∈ / I P ⊇ I P\supseteq I P ⊇ I a ∉ P a\notin P a ∈ / P B = R / P B=R/P B = R / P b = a + P ≠ 0 b=a+P\ne0 b = a + P = 0 B B B B B B b b b c B c = 0 cBc=0 c B c = 0 ( c ∗ c ) ∗ ( c ∗ c ) = c ∗ ( c c ∗ ) c = 0 (c^*c)^*(c^*c)=c^*(cc^*)c=0 ( c ∗ c ) ∗ ( c ∗ c ) = c ∗ ( c c ∗ ) c = 0 c = 0 c=0 c = 0
Suppose that U , V U,V U , V B B B ( V U ) 2 ⊆ V ( U V ) U = 0 (VU)^2\subseteq V(UV)U=0 ( V U ) 2 ⊆ V ( U V ) U = 0 V U = 0 VU=0 V U = 0 T = Ann B ( U ) T=\operatorname{Ann}_B(U) T = Ann B ( U ) S = Ann B ( T ) S=\operatorname{Ann}_B(T) S = Ann B ( T ) B B B T T T S S S V ⊆ T V\subseteq T V ⊆ T T U = 0 TU=0 T U = 0 ( U T ) 2 ⊆ U ( T U ) T = 0 (UT)^2\subseteq U(TU)T=0 ( U T ) 2 ⊆ U ( T U ) T = 0 U T = 0 UT=0 U T = 0 U ⊆ S U\subseteq S U ⊆ S S S S T T T b b b S = S ∗ S=S^* S = S ∗ b ∗ b ∈ S T = 0 b^*b\in ST=0 b ∗ b ∈ S T = 0 b = 0 b=0 b = 0 P P P I I I a a a
Remark 3.2 (Related results). For Noetherian complex ∗ * ∗ [35 Theorem 28] prove the finite-intersection version of Theorem 3.1 . For a finite-dimensional nilpotent real Lie algebra g \mathfrak{g} g U ∗ ( g ) = U ( g ) ⊗ R C U^*(\mathfrak{g})=U(\mathfrak{g})\otimes_{\mathbb{R}}\mathbb{C} U ∗ ( g ) = U ( g ) ⊗ R C x ∗ = − x x^*=-x x ∗ = − x x ∈ g x\in\mathfrak{g} x ∈ g [35 Theorem 7 and Remark 33] expresses every real two-sided ideal of U ∗ ( g ) U^*(\mathfrak{g}) U ∗ ( g ) ∗ * ∗
The Jordan plane
Theorem 3.1 does not extend to left ideals, even in a Noetherian domain. The following example also distinguishes weak reality from reality, answering [4 Remark 5] negatively.
Example 3.3 (The Jordan plane). Let
A = R ⟨ x , y ⟩ / ( y x − x y − x 2 ) , x ∗ = x , y ∗ = − y , A=\mathbb{R}\langle x,y\rangle/(yx-xy-x^2),\qquad x^*=x,\qquad y^*=-y, A = R ⟨ x , y ⟩ / ( y x − x y − x 2 ) , x ∗ = x , y ∗ = − y , and put b = y + x 2 b=y+x^2 b = y + x 2 P = A b P=Ab P = A b I = A x b I=Axb I = A x b A A A
P P P
I I I I I I x x x x ∉ I x\notin I x ∈ / I I I I
Proof. We use a polynomial model and an integral pairing to establish the reality assertions, and normality of x x x
Structure. The Ore extension A = R [ x ] [ y ; δ ] A=\mathbb{R}[x][y;\delta] A = R [ x ] [ y ; δ ] δ = x 2 d / d x \delta=x^2\,d/dx δ = x 2 d / d x ∗ * ∗ y x = x ( y + x ) yx=x(y+x) y x = x ( y + x ) x x x ( x ) = A x = x A (x)=Ax=xA ( x ) = A x = x A A / ( x ) ≅ R [ y ] A/(x)\cong\mathbb{R}[y] A / ( x ) ≅ R [ y ] y ∗ = − y y^*=-y y ∗ = − y ∑ i a ˉ i ( − y ) a ˉ i ( y ) = 0 \sum_i \bar{a}_i(-y)\bar{a}_i(y)=0 ∑ i a ˉ i ( − y ) a ˉ i ( y ) = 0 ∑ i ∣ a ˉ i ( i s ) ∣ 2 = 0 \sum_i|\bar{a}_i(is)|^2=0 ∑ i ∣ a ˉ i ( i s ) ∣ 2 = 0 ( x ) (x) ( x )
A model of A / P A/P A / P The operators
X p = t p , Y p = t 2 p ′ − t 2 p Xp=tp,\qquad Yp=t^2p'-t^2p X p = tp , Y p = t 2 p ′ − t 2 p on R [ t ] \mathbb{R}[t] R [ t ] [ Y , X ] = X 2 [Y,X]=X^2 [ Y , X ] = X 2 b ⋅ 1 = 0 b\cdot1=0 b ⋅ 1 = 0 b b b y y y A A A P P P x x x a + P ↦ a ⋅ 1 a+P\mapsto a\cdot1 a + P ↦ a ⋅ 1 A / P A/P A / P R [ t ] \mathbb{R}[t] R [ t ] x ∉ P x\notin P x ∈ / P
Moreover I = P ∩ ( x ) I=P\cap(x) I = P ∩ ( x ) I ⊆ P ∩ x A I\subseteq P\cap xA I ⊆ P ∩ x A u b ∈ ( x ) ub\in(x) u b ∈ ( x ) ( x ) (x) ( x ) u ˉ y = 0 \bar{u}y=0 u ˉ y = 0 R [ y ] \mathbb{R}[y] R [ y ] u = u ′ x u=u'x u = u ′ x u b = u ′ x b ∈ I ub=u'xb\in I u b = u ′ x b ∈ I x ∉ I x\notin I x ∈ / I
The pairing used to prove reality. Let ν ( f ) \nu(f) ν ( f ) f ∈ R [ t ] f\in\mathbb{R}[t] f ∈ R [ t ] 0 0 0 ν ( 0 ) = ∞ \nu(0)=\infty ν ( 0 ) = ∞ X X X Y Y Y
⟨ f , g ⟩ = ∫ 0 ∞ f ( t ) g ( t ) t − 2 e − 2 t d t . \langle f,g\rangle=\int_0^\infty f(t)g(t)t^{-2}e^{-2t}\,dt. ⟨ f , g ⟩ = ∫ 0 ∞ f ( t ) g ( t ) t − 2 e − 2 t d t . If ν ( f ) + ν ( g ) ≥ 1 \nu(f)+\nu(g)\geq1 ν ( f ) + ν ( g ) ≥ 1 w w w X , Y X,Y X , Y
⟨ w ∗ f , g ⟩ = ⟨ f , w g ⟩ (2) \langle w^*f,g\rangle=\langle f,wg\rangle \tag*{(2)} ⟨ w ∗ f , g ⟩ = ⟨ f , w g ⟩ (2) converge and are equal. For X X X Y Y Y t 2 ⋅ t − 2 e − 2 t = e − 2 t t^2\cdot t^{-2}e^{-2t}=e^{-2t} t 2 ⋅ t − 2 e − 2 t = e − 2 t ⟨ Y f , g ⟩ + ⟨ f , Y g ⟩ = [ f g e − 2 t ] 0 ∞ = 0 \langle Yf,g\rangle+\langle f,Yg\rangle=[fge^{-2t}]_0^\infty=0 ⟨ Y f , g ⟩ + ⟨ f , Y g ⟩ = [ f g e − 2 t ] 0 ∞ = 0 w w w ν \nu ν w w w A A A
Weak reality and failure of reality. Since b + b ∗ = 2 x 2 = x ∗ x + x ∗ x ∈ P + P ∗ b+b^*=2x^2=x^*x+x^*x\in P+P^* b + b ∗ = 2 x 2 = x ∗ x + x ∗ x ∈ P + P ∗ x ∉ P x\notin P x ∈ / P P P P ∑ i a i ∗ a i ∈ P \sum_i a_i^*a_i\in P ∑ i a i ∗ a i ∈ P p i = a i ∗ 1 p_i=a_i^*1 p i = a i ∗ 1 ∑ i a i ∗ p i = 0 \sum_i a_i^*p_i=0 ∑ i a i ∗ p i = 0 x , y ↦ 0 x,y\mapsto0 x , y ↦ 0 P = A b P=Ab P = A b ∑ i c i 2 = 0 \sum_i c_i^2=0 ∑ i c i 2 = 0 c i c_i c i a i a_i a i c i = 0 c_i=0 c i = 0 a i a_i a i ν ( p i ) ≥ 1 \nu(p_i)\geq1 ν ( p i ) ≥ 1 ( f , g ) = ( p i , 1 ) (f,g)=(p_i,1) ( f , g ) = ( p i , 1 )
0 = ∑ i ⟨ a i ∗ p i , 1 ⟩ = ∑ i ∫ 0 ∞ p i ( t ) 2 t − 2 e − 2 t d t . 0=\sum_i\langle a_i^*p_i,1\rangle=\sum_i\int_0^\infty p_i(t)^2t^{-2}e^{-2t}\,dt. 0 = i ∑ ⟨ a i ∗ p i , 1 ⟩ = i ∑ ∫ 0 ∞ p i ( t ) 2 t − 2 e − 2 t d t . Hence every p i = 0 p_i=0 p i = 0 a i ∈ P a_i\in P a i ∈ P P P P
Primeness. Every nonzero two-sided ideal H H H A A A x x x H H H y y y x x x y y y H ∩ R [ x ] = ( f ) H \cap\mathbb{R}[x] = (f) H ∩ R [ x ] = ( f ) δ = x 2 d / d x \delta= x^2d/dx δ = x 2 d / d x f f f x 2 f ′ x^2f' x 2 f ′ f f f x x x x x x A / P ≅ R [ t ] A/P \cong\mathbb{R}[t] A / P ≅ R [ t ] x x x A / P A/P A / P P P P
(2), reality. Suppose S = ∑ i a i ∗ a i ∈ I + I ∗ S = \sum_i a_i^*a_i \in I + I^* S = ∑ i a i ∗ a i ∈ I + I ∗ I + I ∗ ⊆ ( x ) I + I^* \subseteq(x) I + I ∗ ⊆ ( x ) ( x ) (x) ( x ) a i ∈ ( x ) a_i \in(x) a i ∈ ( x ) ν ( p i ) ≥ 1 \nu(p_i) \ge1 ν ( p i ) ≥ 1 p i = a i ⋅ 1 p_i = a_i \cdot1 p i = a i ⋅ 1 a i a_i a i S = u x b + b ∗ x v ∗ S = uxb+b^*xv^* S = ux b + b ∗ x v ∗ h = x v ∗ ⋅ 1 h = xv^* \cdot1 h = x v ∗ ⋅ 1 ν ( h ) ≥ 1 \nu(h) \ge1 ν ( h ) ≥ 1 S ⋅ 1 = b ∗ ⋅ h S \cdot1 = b^* \cdot h S ⋅ 1 = b ∗ ⋅ h b ⋅ 1 = 0 b \cdot1 = 0 b ⋅ 1 = 0
∑ i ∫ 0 ∞ p i 2 t − 2 e − 2 t d t = ∑ i ⟨ a i ∗ p i , 1 ⟩ = ⟨ b ∗ h , 1 ⟩ = ⟨ h , b ⋅ 1 ⟩ = 0. \sum_i \int_0^\infty p_i^2t^{-2}e^{-2t}\,dt = \sum_i \langle a_i^*p_i,1\rangle= \langle b^*h,1\rangle= \langle h,b\cdot1\rangle= 0. i ∑ ∫ 0 ∞ p i 2 t − 2 e − 2 t d t = i ∑ ⟨ a i ∗ p i , 1 ⟩ = ⟨ b ∗ h , 1 ⟩ = ⟨ h , b ⋅ 1 ⟩ = 0. Thus every p i = 0 p_i = 0 p i = 0 a i ∈ P ∩ ( x ) = I a_i \in P \cap(x) = I a i ∈ P ∩ ( x ) = I I I I
(2), separation. Let J ⊇ I J \supseteq I J ⊇ I H = A / J H = A/J H = A / J q = 1 + J ≠ 0 q = 1 + J \ne0 q = 1 + J = 0 K = ker ( x : H → H ) K = \ker(x:H \to H) K = ker ( x : H → H ) K = 0 K = 0 K = 0 x b q = 0 xbq = 0 x b q = 0 b q = 0 bq = 0 b q = 0 b ∈ J b \in J b ∈ J x ∗ x + x x ∗ = b + b ∗ ∈ J + J ∗ x^*x + x x^* = b+b^* \in J+J^* x ∗ x + x x ∗ = b + b ∗ ∈ J + J ∗ x ∈ J x \in J x ∈ J x q = 0 xq = 0 x q = 0 q ∈ K = 0 q \in K = 0 q ∈ K = 0 K ≠ 0 K \ne0 K = 0 x ∈ Ann A ( K ) = Ann A ( H ) x \in\operatorname{Ann}_A(K) = \operatorname{Ann}_A(H) x ∈ Ann A ( K ) = Ann A ( H ) ( x ) ⊆ J (x) \subseteq J ( x ) ⊆ J
Hereditary Noetherian prime rings
The Jordan plane of Example 3.3 is the Ore extension A = R [ x ] [ y ; x 2 d / d x ] A = \mathbb{R}[x][y;x^2d/dx] A = R [ x ] [ y ; x 2 d / d x ] ( x ) (x) ( x ) A / ( x ) ≅ R [ y ] A/(x) \cong\mathbb{R}[y] A / ( x ) ≅ R [ y ]
We prove the separation statement for arbitrary modules, beginning with the definitions and basic properties needed for the argument.
For a left module Q Q Q Q ‾ \overline{Q} Q q ‾ a = a ∗ q \overline{q}a = a^*q q a = a ∗ q Q ‾ ⊗ R Q \overline{Q} \otimes_R Q Q ⊗ R Q
( q ‾ a ) ⊗ v = q ‾ ⊗ ( a v ) ( a ∈ R ) . (\overline{q}a) \otimes v = \overline{q} \otimes(av) \qquad(a \in R). ( q a ) ⊗ v = q ⊗ ( a v ) ( a ∈ R ) . Definition 3.4 (Real and weakly real modules). An R R R Q Q Q real if, for every finite family q i ∈ Q q_i \in Q q i ∈ Q
∑ i q i ‾ ⊗ q i = 0 in Q ‾ ⊗ R Q ⟹ q i = 0 for all i , \sum_i \overline{q_i} \otimes q_i = 0 \quad\text{in } \overline{Q} \otimes_R Q \quad\Longrightarrow\quad q_i = 0 \text{ for all } i, i ∑ q i ⊗ q i = 0 in Q ⊗ R Q ⟹ q i = 0 for all i , and weakly real if, for every q ∈ Q q \in Q q ∈ Q a i ∈ R a_i \in R a i ∈ R
( ∑ i a i ∗ a i ) q = 0 ⟹ a i q = 0 for all i . \left(\sum_i a_i^*a_i\right)q = 0 \quad\Longrightarrow\quad a_iq = 0 \text{ for all } i. ( i ∑ a i ∗ a i ) q = 0 ⟹ a i q = 0 for all i . A submodule N ⊆ M N \subseteq M N ⊆ M real or weakly real if M / N M/N M / N real prime if M / N M/N M / N
Every real module is weakly real, by the identity
∑ i a i q ‾ ⊗ a i q = q ‾ ⊗ ( ∑ i a i ∗ a i ) q . \sum_i \overline{a_iq} \otimes a_iq = \overline{q} \otimes\left(\sum_i a_i^*a_i\right)q. i ∑ a i q ⊗ a i q = q ⊗ ( i ∑ a i ∗ a i ) q . The next proposition checks that these module definitions recover the definitions for left ideals.
Proposition 3.5 (Cyclic real modules). Let I I I R R R
(1) I I I R / I R/I R / I
(2) I I I R / I R/I R / I I I I R / I R/I R / I
Proof. (1) The map a ‾ ⊗ b ↦ a ∗ b \overline{a}\otimes b\mapsto a^*b a ⊗ b ↦ a ∗ b R / I ‾ ⊗ R R / I ≅ R / ( I ∗ + I ) \overline{R/I}\otimes_R R/I\cong R/(I^*+I) R / I ⊗ R R / I ≅ R / ( I ∗ + I ) c + ( I ∗ + I ) ↦ 1 + I ⊗ ( c + I ) c+(I^*+I)\mapsto1+I\otimes(c+I) c + ( I ∗ + I ) ↦ 1 + I ⊗ ( c + I ) I I I I ∗ I^* I ∗ i ∈ I i\in I i ∈ I
1 + I ‾ ⊗ ( i ∗ + I ) = i ‾ + I ⊗ ( 1 + I ) = 0. \overline{1+I}\otimes(i^*+I)=\overline{i}+I\otimes(1+I)=0. 1 + I ⊗ ( i ∗ + I ) = i + I ⊗ ( 1 + I ) = 0. The balancing relation shows that the two maps are inverse. The isomorphism sends ∑ i a i ‾ ⊗ a i \sum_i\overline{a_i}\otimes a_i ∑ i a i ⊗ a i ∑ i a i ∗ a i + ( I + I ∗ ) \sum_i a_i^*a_i+(I+I^*) ∑ i a i ∗ a i + ( I + I ∗ )
(2) If I I I ( ∑ i a i ∗ a i ) b ∈ I (\sum_i a_i^*a_i)b\in I ( ∑ i a i ∗ a i ) b ∈ I b ∗ b^* b ∗ ∑ i ( a i b ) ∗ ( a i b ) ∈ I \sum_i(a_ib)^*(a_ib)\in I ∑ i ( a i b ) ∗ ( a i b ) ∈ I a i b ∈ I a_ib\in I a i b ∈ I b = 1 b=1 b = 1 a R a ⊆ I aRa\subseteq I a R a ⊆ I ( a ∗ a ) ∗ ( a ∗ a ) = a ∗ ( a a ∗ a ) ∈ I (a^*a)^*(a^*a)=a^*(aa^*a)\in I ( a ∗ a ) ∗ ( a ∗ a ) = a ∗ ( a a ∗ a ) ∈ I a ∗ a ∈ I a^*a\in I a ∗ a ∈ I a ∈ I a\in I a ∈ I I I I R / I R/I R / I Lemma 2.2 (4).
The following closure and annihilator properties will be used in the separation arguments.
Lemma 3.6 (Basic properties). Real and weakly real modules are closed under submodules, products and restriction of scalars along ∗ * ∗
Proof. For real modules, the closure assertions follow by mapping tensor relations into the ambient tensor product, into each coordinate, or along the ring map. For weakly real modules they follow directly from (3). Weak reality of real modules was noted above. Let Q Q Q T = Ann R ( L ) T=\operatorname{Ann}_R(L) T = Ann R ( L ) L L L a ∈ T a\in T a ∈ T a ∗ L ⊆ L a^*L\subseteq L a ∗ L ⊆ L a a ∗ L = 0 aa^*L=0 a a ∗ L = 0 a ∗ L = 0 a^*L=0 a ∗ L = 0 T = T ∗ T=T^* T = T ∗ L L L T T T Q Q Q z = ∑ j t j q j ∈ L ∩ T Q z=\sum_jt_jq_j\in L\cap TQ z = ∑ j t j q j ∈ L ∩ T Q z ‾ ⊗ z = ∑ j t j ∗ z ‾ ⊗ q j = 0 \overline{z}\otimes z=\sum_j\overline{t_j^*z}\otimes q_j=0 z ⊗ z = ∑ j t j ∗ z ⊗ q j = 0 z = 0 z=0 z = 0 Q Q Q
Intersections of real submodules are real, since the quotient embeds in the product of the real quotients. Their intersection defines the real radical N r e \sqrt[\mathrm{re}]{N} re N N N N
The proof uses only the Artinian quotient property for nonzero real ideals, as the following criterion shows.
Theorem 3.8 (Real prime separation from Artinian factors). Let R R R R / T R/T R / T T T T R R R
Proof. It suffices to separate each nonzero element q q q M M M P P P q ∉ P q\notin P q ∈ / P Q = M / P Q=M/P Q = M / P Q = Q 1 ⊕ Q 2 Q=Q_1\oplus Q_2 Q = Q 1 ⊕ Q 2 Q i Q_i Q i Q Q Q Q / Q i Q/Q_i Q / Q i Q Q Q Q i Q_i Q i q + P q+P q + P P P P Q Q Q
If Q Q Q 0 ≠ H ⊆ Q 0\ne H\subseteq Q 0 = H ⊆ Q T = Ann R ( H ) ⊋ Ann R ( Q ) T=\operatorname{Ann}_R(H)\supsetneq\operatorname{Ann}_R(Q) T = Ann R ( H ) ⊋ Ann R ( Q ) T ≠ 0 T\ne0 T = 0 Lemma 3.6 , T T T R / T R/T R / T Q Q Q H ∩ T Q = 0 H\cap TQ=0 H ∩ T Q = 0 H H H Q / T Q Q/TQ Q / T Q R / T R/T R / T Q → Q / T Q Q\to Q/TQ Q → Q / T Q H H H Q Q Q Q Q Q P P P q q q
:::
Every quotient of a hereditary Noetherian prime ring by a nonzero two-sided ideal is Artinian [37 Lemma 2(1)] . The theorem therefore proves Theorem B(3), including the case of noncommutative Dedekind prime rings. In particular, D [ x ; σ , δ ] D[x;\sigma,\delta] D [ x ; σ , δ ] D D D σ \sigma σ δ \delta δ σ \sigma σ x a = σ ( a ) x + δ ( a ) xa=\sigma(a)x+\delta(a) x a = σ ( a ) x + δ ( a ) a ∈ D a\in D a ∈ D
Remark 3.9 (Localizing the Jordan plane). Passing to division coefficients does not preserve the real left ideal in Example 3.3. Indeed, the derivation and involution extend to
A ~ = R ( x ) [ y ; x 2 d / d x ] , x ∗ = x , y ∗ = − y , \widetilde{A}=\mathbb{R}(x)[y;x^{2}d/dx],\qquad x^{*}=x,\qquad y^{*}=-y, A = R ( x ) [ y ; x 2 d / d x ] , x ∗ = x , y ∗ = − y , so Theorem 3.8 applies to A ~ \widetilde{A} A x x x I = A x b I=Ax b I = A x b A ~ I ~ = A ~ b \widetilde{A}\widetilde{I}=\widetilde{A}b A I = A b b = y + x 2 b=y+x^{2} b = y + x 2 y y y
b + b ∗ = 2 x 2 = x ∗ x + x ∗ x ∈ A ~ b + ( A ~ b ) ∗ . b+b^{*}=2x^{2}=x^{*}x+x^{*}x\in\widetilde{A}b+(\widetilde{A}b)^{*}. b + b ∗ = 2 x 2 = x ∗ x + x ∗ x ∈ A b + ( A b ) ∗ . Reality would force the unit x x x A ~ b \widetilde{A}b A b I I I A A A
Hereditary Noetherian prime rings need not be real Jacobson, even when they are Jacobson.
Example 3.10 (The integers). With the identity involution, Z \mathbb{Z} Z ( 0 ) = ⋂ p p r i m e p Z (0)=\bigcap_{p\ \mathrm{prime}}p\mathbb{Z} ( 0 ) = ⋂ p prime p Z p Z p\mathbb{Z} p Z p p p 1 ∗ 1 1^{*}1 1 ∗ 1 1 ∉ p Z 1\notin p\mathbb{Z} 1 ∈ / p Z ( 0 ) (0) ( 0 ) Z \mathbb{Z} Z
Semireality and real maximal left ideals
Write Σ R ∗ R \Sigma R^{*}R Σ R ∗ R a ∗ a a^{*}a a ∗ a ∗ * ∗ R R R semireal if − 1 ∉ Σ R ∗ R -1\notin\Sigma R^{*}R − 1 ∈ / Σ R ∗ R
Recall that an element of a ring B B B regular if it is neither a left nor a right zero divisor. When the set S S S S − 1 B S^{-1}B S − 1 B classical left quotient ring , or classical left localization , of B B B s − 1 b s^{-1}b s − 1 b s ∈ S s\in S s ∈ S b ∈ B b\in B b ∈ B B B B [28 2.3.6] . Every semisimple Artinian ring Q Q Q a ∈ Q a\in Q a ∈ Q b ∈ Q b\in Q b ∈ Q a b a = a aba=a aba = a
Theorem 3.11 (Semireality). Let R R R ∗ * ∗ R / J R/J R / J J J J ∗ * ∗ [28 Chapter 2] , or a PI ∗ * ∗ [13 Part B, Section 6] ). The following conditions are equivalent:
R R R
R R R
R R R
R R R
There exists a nonzero real left R R R
These conditions need not imply the existence of a real maximal left ideal, even when R R R
Proof. We first prove (1)⇒ \Rightarrow ⇒ ∗ * ∗ B B B Sym ( B ) = { b : b ∗ = b } \operatorname{Sym}(B)=\{b:b^{*}=b\} Sym ( B ) = { b : b ∗ = b } 1 1 1 m ↦ x ∗ m x m\mapsto x^{*}mx m ↦ x ∗ m x x ∈ B x\in B x ∈ B − 1 -1 − 1 Σ R ∗ R \Sigma R^{*}R Σ R ∗ R M M M [3 Theorem 1] , there is a proper involution-invariant prime two-sided ideal J J J
J ∩ Sym ( R ) = M ∩ ( − M ) . J\cap\operatorname{Sym}(R)=M\cap(-M). J ∩ Sym ( R ) = M ∩ ( − M ) . Its image M ‾ \overline{M} M B = R / J B=R/J B = R / J M ‾ ∩ ( − M ‾ ) = 0 \overline{M}\cap(-\overline{M})=0 M ∩ ( − M ) = 0 m , n ∈ M m,n\in M m , n ∈ M m + n ∈ M ∩ ( − M ) m+n\in M\cap(-M) m + n ∈ M ∩ ( − M ) − m = n − ( m + n ) ∈ M -m=n-(m+n)\in M − m = n − ( m + n ) ∈ M m ∈ J m\in J m ∈ J − 1 -1 − 1 m ∈ M m\in M m ∈ M 1 + m ∈ J ∩ Sym ( R ) = M ∩ ( − M ) 1+m\in J\cap\operatorname{Sym}(R)=M\cap(-M) 1 + m ∈ J ∩ Sym ( R ) = M ∩ ( − M ) − 1 = m − ( 1 + m ) ∈ M -1=m-(1+m)\in M − 1 = m − ( 1 + m ) ∈ M
By hypothesis B B B Q Q Q Q Q Q [3 Proposition 2] ,
{ q ∈ Sym ( Q ) : c ∗ q c ∈ M ‾ for some regular c ∈ B } \{q\in\operatorname{Sym}(Q):c^*qc\in\overline{M}\text{ for some regular }c\in B\} { q ∈ Sym ( Q ) : c ∗ q c ∈ M for some regular c ∈ B } is a proper quadratic module on Q Q Q P P P [3 Theorem 1] , P ∩ ( − P ) P\cap(-P) P ∩ ( − P ) Q Q Q P ∩ ( − P ) = 0 P\cap(-P)=0 P ∩ ( − P ) = 0
We show that Q Q Q ∑ i a i ∗ a i = 0 \sum_i a_i^*a_i=0 ∑ i a i ∗ a i = 0 a i ∗ a i a_i^*a_i a i ∗ a i P ∩ ( − P ) P\cap(-P) P ∩ ( − P ) a = a i a=a_i a = a i b ∈ Q b\in Q b ∈ Q a b a = a aba=a aba = a p = a b p=ab p = ab e = p + p ∗ − p p ∗ e=p+p^*-pp^* e = p + p ∗ − p p ∗ p 2 = p p^2=p p 2 = p p ∗ p = 0 p^*p=0 p ∗ p = 0 a = p a a=pa a = p a e ∗ = e e^*=e e ∗ = e e 2 = e e^2=e e 2 = e e p = p ep=p e p = p
e = e ∗ e ∈ P , − e = ( p − p ∗ ) ∗ ( p − p ∗ ) ∈ P . e=e^*e\in P,\qquad-e=(p-p^*)^*(p-p^*)\in P. e = e ∗ e ∈ P , − e = ( p − p ∗ ) ∗ ( p − p ∗ ) ∈ P . Thus e = 0 e=0 e = 0 p = e p = 0 p=ep=0 p = e p = 0 a = p a = 0 a=pa=0 a = p a = 0 Q Q Q ∗ * ∗ B B B J J J
The implications (2)⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ I I I − 1 = ∑ i a i ∗ a i -1=\sum_i a_i^*a_i − 1 = ∑ i a i ∗ a i 1 ∗ 1 + ∑ i a i ∗ a i = 0 ∈ I 1^*1+\sum_i a_i^*a_i=0\in I 1 ∗ 1 + ∑ i a i ∗ a i = 0 ∈ I 1 ∈ I 1\in I 1 ∈ I ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒
Only (1)⇒ \Rightarrow ⇒
To study real maximal left ideals, we first establish the relevant facts about semisimple modules and weak reality.
Proposition 3.12 (Reality and semisimplicity). Let R R R ∗ * ∗
(1) If R R R R R R
(2) A semisimple left R R R
Consequently, a maximal left ideal, or a finite intersection of maximal left ideals, is real if and only if it is weakly real. This equivalence can fail for infinite intersections (Example 6.6). Here maximal means maximal among all proper left ideals.
Proof. (1) The regular module of R R R U U U 0 ≠ u ∈ U 0\ne u\in U 0 = u ∈ U π : R → U \pi:R\to U π : R → U a ↦ a u a\mapsto au a ↦ a u R = ker π ⊕ L R=\ker\pi\oplus L R = ker π ⊕ L U ≅ L ⊆ R U\cong L\subseteq R U ≅ L ⊆ R
(2) Only the implication from weak reality to reality needs proof. First let L L L ∑ i a i ∗ a i = ℓ + m ∗ \sum_i a_i^*a_i=\ell+m^* ∑ i a i ∗ a i = ℓ + m ∗ ℓ , m ∈ L \ell,m\in L ℓ , m ∈ L a j ∉ L a_j\notin L a j ∈ / L r ∈ R r\in R r ∈ R 1 − r a j ∈ L 1-ra_j\in L 1 − r a j ∈ L b i = a i b_i=a_i b i = a i i ≠ j i\ne j i = j b j = a j − r ∗ m b_j=a_j-r^*m b j = a j − r ∗ m
∑ i b i ∗ b i = ℓ + m ∗ ( 1 − r a j ) − a j ∗ r ∗ m + m ∗ r r ∗ m ∈ L . \sum_i b_i^*b_i=\ell+m^*(1-ra_j)-a_j^*r^*m+m^*rr^*m\in L. i ∑ b i ∗ b i = ℓ + m ∗ ( 1 − r a j ) − a j ∗ r ∗ m + m ∗ r r ∗ m ∈ L . Weak reality gives b j ∈ L b_j\in L b j ∈ L a j = b j + r ∗ m ∈ L a_j=b_j+r^*m\in L a j = b j + r ∗ m ∈ L L L L I = ⋂ j = 1 n L j I=\bigcap_{j=1}^{n}L_j I = ⋂ j = 1 n L j L j L_j L j R / I R/I R / I ⨁ j = 1 n R / L j \bigoplus_{j=1}^{n}R/L_j ⨁ j = 1 n R / L j
If M M M R / Ann R ( M ) R/\operatorname{Ann}_{R}(M) R / Ann R ( M ) Lemma 3.6 , so Proposition 3.12 (1) also gives reality of M M M
The next proposition characterizes the existence of real maximal left ideals over an arbitrary unital ∗ * ∗ ∗ * ∗
Proposition 3.13 (Real simple modules and representations). Let R R R ∗ * ∗
R R R
R R R
R R R
R R R ∗ * ∗
In (4), the inner product is biadditive and Hermitian, satisfies ⟨ a u , v ⟩ = ⟨ u , a ∗ v ⟩ \langle au,v\rangle=\langle u,a^{*}v\rangle ⟨ a u , v ⟩ = ⟨ u , a ∗ v ⟩ ∑ i ⟨ u i , u i ⟩ = 0 \sum_i\langle u_i,u_i\rangle=0 ∑ i ⟨ u i , u i ⟩ = 0 u i = 0 u_i=0 u i = 0
core ( L ) = { a ∈ R : a R ⊆ L } = Ann R ( R / L ) \operatorname{core}(L)=\{a\in R:aR\subseteq L\}=\operatorname{Ann}_{R}(R/L) core ( L ) = { a ∈ R : a R ⊆ L } = Ann R ( R / L ) of real maximal left ideals L L L R R R Proposition 3.12 , “real” may be replaced by “weakly real” in (1) and (2).
Proof. For a real simple module S S S 0 ≠ s ∈ S 0\ne s\in S 0 = s ∈ S L s = Ann R ( s ) L_s=\operatorname{Ann}_{R}(s) L s = Ann R ( s ) Proposition 3.5 , since R / L s ≅ S R/L_s\cong S R / L s ≅ S R s = S Rs=S R s = S
Ann R ( S ) = core ( L s ) = ⋂ 0 ≠ t ∈ S Ann R ( t ) , \operatorname{Ann}_{R}(S)=\operatorname{core}(L_s)=\bigcap_{0\ne t\in S}\operatorname{Ann}_{R}(t), Ann R ( S ) = core ( L s ) = 0 = t ∈ S ⋂ Ann R ( t ) , an intersection of real maximal left ideals. Conversely, a real maximal left ideal L L L R / L R/L R / L Lemma 3.6 , and primitive by definition. A proper real left ideal excludes − 1 ∈ Σ R ∗ R -1\in\Sigma R^{*}R − 1 ∈ Σ R ∗ R Theorem 3.11 , so these conditions imply semireality. For (2)⇔ \Leftrightarrow ⇔ S S S
⟨ u , v ⟩ = u ˉ ⊗ v in S ˉ ⊗ R S , ⟨ a u , v ⟩ = ⟨ u , a ∗ v ⟩ . \langle u,v\rangle=\bar{u}\otimes v\text{ in }\bar{S}\otimes_{R}S,\qquad\langle au,v\rangle=\langle u,a^{*}v\rangle. ⟨ u , v ⟩ = u ˉ ⊗ v in S ˉ ⊗ R S , ⟨ a u , v ⟩ = ⟨ u , a ∗ v ⟩ . The target has involution ( u ˉ ⊗ v ) ∗ = v ˉ ⊗ u (\bar{u}\otimes v)^{*}=\bar{v}\otimes u ( u ˉ ⊗ v ) ∗ = v ˉ ⊗ u
PI rings
This section proves Theorem C. Central localization gives prime and real prime separation for arbitrary modules, and an embedding criterion transfers the (real) Jacobson property from rings to finitely generated modules. The real Jacobson property can fail for formally real affine PI algebras. We introduce positive real prime ideals to study the existence of real maximal left ideals and give sufficient criteria for the real Jacobson property.
Prime and real prime separation
Here a P I PI P I Z \mathbb{Z} Z 1 1 1
Definition 4.1 (Real prime-separated modules). For a ring with involution, a left R R R R R R
A semiprime PI ring need not be Goldie (e.g. ∏ m ≥ 1 R \prod_{m\ge1}\mathbb{R} ∏ m ≥ 1 R
Lemma 4.2 (Central localization). Let B B B C C C 0 ≠ c ∈ C 0 \ne c \in C 0 = c ∈ C p \mathfrak{p} p C C C c c c S = C ∖ p S = C \setminus\mathfrak{p} S = C ∖ p
that is finite-dimensional central simple over C p C_{\mathfrak{p}} C p B B B
Proof. By [34 Theorem 6.1.28, Corollary 6.1.22] , B B B n n n g n g_n g n B B B c ≠ 0 c \ne0 c = 0 B [ c − 1 ] B[c^{-1}] B [ c − 1 ] n 2 n^2 n 2 C [ c − 1 ] C[c^{-1}] C [ c − 1 ] [34 Corollary 6.1.36] , so K ⊗ C [ c − 1 ] B [ c − 1 ] K \otimes_{C[c^{-1}]} B[c^{-1}] K ⊗ C [ c − 1 ] B [ c − 1 ] n 2 n^2 n 2 C [ c − 1 ] → K C[c^{-1}] \to K C [ c − 1 ] → K [13 Part B, Section 8] . The centre C C C z ∈ C z \in C z ∈ C C C C c ≠ 0 c \ne0 c = 0 c c c p \mathfrak{p} p C p C_{\mathfrak{p}} C p c ∈ S c \in S c ∈ S
A = S − 1 B ≅ C p ⊗ C [ c − 1 ] B [ c − 1 ] A = S^{-1}B \cong C_{\mathfrak{p}} \otimes_{C[c^{-1}]} B[c^{-1}] A = S − 1 B ≅ C p ⊗ C [ c − 1 ] B [ c − 1 ] is central simple of dimension n 2 n^2 n 2 B → A B \to A B → A
Suppose B B B z ∈ p z \in\mathfrak{p} z ∈ p u z = 0 uz = 0 u z = 0 u ∉ p u \notin\mathfrak{p} u ∈ / p p C p = 0 \mathfrak{p}C_{\mathfrak{p}} = 0 p C p = 0 ( u z ∗ ) ( u z ∗ ) ∗ = ( u z ) ( u z ∗ ) = 0 (uz^*)(uz^*)^* = (uz)(uz^*) = 0 ( u z ∗ ) ( u z ∗ ) ∗ = ( u z ) ( u z ∗ ) = 0 u z ∗ = 0 uz^* = 0 u z ∗ = 0 z ∗ ∈ p z^* \in\mathfrak{p} z ∗ ∈ p S ∗ = S S^* = S S ∗ = S A A A ∑ i ( b i / s ) ∗ ( b i / s ) = 0 \sum_i (b_i/s)^*(b_i/s) = 0 ∑ i ( b i / s ) ∗ ( b i / s ) = 0 t ∑ i b i ∗ b i = 0 t\sum_i b_i^*b_i = 0 t ∑ i b i ∗ b i = 0 t ∈ S t \in S t ∈ S ∑ i ( t b i ) ∗ ( t b i ) = 0 \sum_i (tb_i)^*(tb_i) = 0 ∑ i ( t b i ) ∗ ( t b i ) = 0 t b i = 0 tb_i = 0 t b i = 0 b i / s = 0 b_i/s = 0 b i / s = 0
Theorem 4.3 (Prime separation for PI rings). Let R R R R R R R R R R R R
Proof. Let M M M 0 ≠ q ∈ M 0 \ne q \in M 0 = q ∈ M L = R q L = Rq L = R q T = Ann R ( L ) T = \operatorname{Ann}_{R}(L) T = Ann R ( L ) B = R / T B = R/T B = R / T B B B L L L B B B L ∩ T M = 0 L \cap TM = 0 L ∩ T M = 0 Lemma 2.5 (2) and semiprimeness of M M M A = S − 1 B A = S^{-1}B A = S − 1 B Lemma 4.2 and put V = S − 1 ( M / T M ) V = S^{-1}(M/TM) V = S − 1 ( M / T M ) M → V M \to V M → V q q q s ~ q ∈ L ∩ T M = 0 \widetilde{s}q \in L \cap TM = 0 s q ∈ L ∩ T M = 0 s ~ \widetilde{s} s s ∈ S s \in S s ∈ S s s s B B B s ~ ( r q ) = r s ~ q = 0 \widetilde{s}(rq) = r\widetilde{s}q = 0 s ( r q ) = r s q = 0 r r r s L = 0 sL = 0 s L = 0
For every nonzero B B B E ⊆ V E \subseteq V E ⊆ V A A A S − 1 E S^{-1}E S − 1 E A A A b E = 0 bE = 0 b E = 0 b / 1 = 0 b/1 = 0 b /1 = 0 ker ( B → A ) \ker(B \to A) ker ( B → A ) V V V Ann B ( E ) = ker ( B → A ) \operatorname{Ann}_{B}(E) = \ker(B \to A) Ann B ( E ) = ker ( B → A ) E E E V V V B B B R R R P = ker ( M → V ) P = \ker(M \to V) P = ker ( M → V ) q q q
If M M M T T T Lemma 3.6 , so A A A V V V R R R Proposition 3.12 (1) and restriction of scalars (Lemma 3.6 ). Hence M / P M/P M / P V V V M M M M M M Lemma 3.6 . Applying the construction to real quotients proves real prime separation, and a weakly real left ideal is real because its quotient is semiprime and weakly real (Proposition 3.5 ). □
Remark 4.4. Theorem 4.3 applies to rings module-finite over their centre, for instance matrix rings over commutative rings. For commutative rings with trivial involution, the characterization of real modules is the annihilator criterion of [7 Proposition 4.3] ; semiprimeness cannot be omitted [7 Example 4.4, 21, Theorem 11] . For left ideals, the theorem extends [4 Corollary 1] to all PI *-rings and answers the commutative nontrivial-involution question in [7 ] . The prime-separation argument uses only simplicity of A A A E ⊗ k C E \otimes_k C E ⊗ k C E E E k k k C C C [15 Introduction and Corollary 3.12] .
Jacobson rings and modules
Definition 4.5 (Jacobson modules). A left R R R M M M Jacobson if every prime submodule is an intersection of maximal submodules. If R R R M M M real Jacobson if every real prime submodule is an intersection of real maximal submodules. Thus R R R
For commutative rings, Jacobson modules in this sense are the Hilbert modules of [27 ] ; our left Jacobson rings are the weakly left Jacobson rings of [10 ] , and their strongly left Jacobson rings are the rings that are both prime-separated and left Jacobson. For a prime-separated module, the Jacobson property means that every semiprime submodule is an intersection of maximal ones, and similarly in the real case.
Definition 4.6 (Prime embedding property). A ring R R R prime embedding property if every finitely generated prime left R R R V V V R R R R / Ann R ( V ) R/\operatorname{Ann}_{R}(V) R / Ann R ( V ) real prime embedding property if this holds for every finitely generated real prime left module.
Theorem 4.7 (Jacobson transfer). Let R R R
(1) If R R R
(a) R R R
(b) R R R
(c) Every finitely generated left R R R
(2) If R R R
(a) R R R
(b) R R R
(c) Every finitely generated left R R R
The implications (c)⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒
Proof. We prove both parts together, imposing reality in (2). (c)⇒ \Rightarrow ⇒ M = R M=R M = R ⇒ \Rightarrow ⇒ T T T T = ⋂ i L i T=\bigcap_i L_i T = ⋂ i L i L i L_i L i
T ⊆ Ann R ( R / L i ) ⊆ L i , T = ⋂ i Ann R ( R / L i ) . T \subseteq\operatorname{Ann}_{R}(R/L_i) \subseteq L_i,\qquad T=\bigcap_i\operatorname{Ann}_{R}(R/L_i). T ⊆ Ann R ( R / L i ) ⊆ L i , T = i ⋂ Ann R ( R / L i ) . These annihilators are primitive, and real primitive in (2).
For (a)⇒ \Rightarrow ⇒ P P P M M M a ∉ P a\notin P a ∈ / P V = M / P V=M/P V = M / P T = Ann R ( V ) T=\operatorname{Ann}_{R}(V) T = Ann R ( V ) T T T Lemma 3.6 . By (a), T T T U U U
Ann R ( U ) = ⋂ 0 ≠ u ∈ U Ann R ( u ) \operatorname{Ann}_{R}(U)=\bigcap_{0\ne u\in U}\operatorname{Ann}_{R}(u) Ann R ( U ) = 0 = u ∈ U ⋂ Ann R ( u ) with R / Ann R ( u ) ≅ U R/\operatorname{Ann}_{R}(u)\cong U R / Ann R ( u ) ≅ U T T T T T T
The embedding property supplies an R R R f : V → R / T f:V\to R/T f : V → R / T f ( a + P ) ≠ 0 f(a+P)\ne0 f ( a + P ) = 0 L ⊇ T L\supseteq T L ⊇ T f ( a + P ) f(a+P) f ( a + P )
M ⟶ V → f R / T ⟶ R / L M\longrightarrow V\xrightarrow{f}R/T\longrightarrow R/L M ⟶ V f R / T ⟶ R / L is nonzero at a a a R / L R/L R / L P P P a a a R / L R/L R / L a a a
The first Weyl algebra in characteristic zero is Jacobson but not left Jacobson [10 Theorem 7] . The real converse can also fail: in separate work we will exhibit a formally real division ∗ * ∗ D D D D [ z ] D[z] D [ z ]
We now verify the prime embedding property for all PI rings.
Lemma 4.8 (Prime embedding for PI rings). Every PI ring R R R R R R V V V R R R ( R / Ann R ( V ) ) r (R/\operatorname{Ann}_{R}(V))^{r} ( R / Ann R ( V ) ) r r r r
Proof. Let V V V B = R / Ann R ( V ) B=R/\operatorname{Ann}_{R}(V) B = R / Ann R ( V ) B B B V V V B B B [31 ] , [13 Part B, Section 6] , Q = S − 1 B Q=S^{-1}B Q = S − 1 B S = Z ( B ) ∖ { 0 } S=Z(B)\setminus\{0\} S = Z ( B ) ∖ { 0 } s v = 0 sv=0 s v = 0 s ∈ S s\in S s ∈ S v ≠ 0 v\ne0 v = 0 s B v = 0 sBv=0 s B v = 0 Ann B ( B v ) = 0 \operatorname{Ann}_{B}(Bv)=0 Ann B ( B v ) = 0 V ↪ S − 1 V V\hookrightarrow S^{-1}V V ↪ S − 1 V Q r ↠ S − 1 V Q^{r}\twoheadrightarrow S^{-1}V Q r ↠ S − 1 V f : V → Q r f:V\to Q^{r} f : V → Q r s s s V V V v ↦ s f ( v ) v\mapsto sf(v) v ↦ s f ( v ) V V V B r B^{r} B r
Theorem 4.7 now gives the Jacobson equivalences in Theorem C.
Corollary 4.9 (Amitsur–Procesi for modules). Let R R R k k k R R R k k k R R R
Proof. The algebra R R R [13 Part B, Theorem 11.7] , so Theorems 4.3 and 4.7 , together with Lemma 4.8 , give the intersection assertions. Let L L L R R R M M M U = M / L U=M/L U = M / L A = R / Ann R ( U ) A=R/\operatorname{Ann}_{R}(U) A = R / Ann R ( U ) F F F A A A [ F : k ] < ∞ [F:k]<\infty [ F : k ] < ∞ A A A U U U k k k L L L k k k [13 Part B, Theorem 11.5] . □
Proposition 4.11 (Real primitive ideals of PI rings). Let R R R J J J R R R R R R k k k R R R k k k
Proof. Real primitive ideals are real and primitive by Proposition 3.13 . Conversely, if J J J J = J ∗ J=J^{*} J = J ∗ R / J R/J R / J [13 Part B, Theorem 4.13] . Its simple modules are real by Proposition 3.12 (1) and faithful, so J J J k k k Corollary 4.9 ). □
A formally real affine PI counterexample
An affine PI algebra need not be real Jacobson, even though it is real prime-separated by Theorem 4.3 .
Example 4.12 (A twisted adjoint). Put
C = R [ x ± 1 , y ± 1 ] , B = M 4 ( C ) , S = diag ( 1 , x , y , − x y ) , C=\mathbb{R}[x^{\pm1},y^{\pm1}],\qquad B=M_{4}(\mathbb{C}),\qquad S=\operatorname{diag}(1,x,y,-xy), C = R [ x ± 1 , y ± 1 ] , B = M 4 ( C ) , S = diag ( 1 , x , y , − x y ) , and define a ∗ = S − 1 a T S a^{*}=S^{-1}a^{\mathsf{T}}S a ∗ = S − 1 a T S B B B R \mathbb{R} R B B B 0 0 0 B B B
Proof. Since S S S C C C R \mathbb{R} R C C C Formal reality. We first observe that
p 1 + x p 2 + y p 3 − x y p 4 = 0 , p k sums of squares in C , p_1+xp_2+yp_3-xyp_4=0,\qquad p_k\text{ sums of squares in }C, p 1 + x p 2 + y p 3 − x y p 4 = 0 , p k sums of squares in C , forces every p k = 0 p_k=0 p k = 0 Z 2 \mathbb{Z}^2 Z 2 2 Z 2 2\mathbb{Z}^2 2 Z 2 ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) (0,0),(1,0),(0,1),(1,1) ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 1 )
Write s 1 = 1 s_1=1 s 1 = 1 s 2 = x s_2=x s 2 = x s 3 = y s_3=y s 3 = y s 4 = − x y s_4=-xy s 4 = − x y ∑ ν a ν ∗ a ν = 0 \sum_\nu a_\nu^*a_\nu=0 ∑ ν a ν ∗ a ν = 0 ( j , j ) (j,j) ( j , j ) s j s_j s j
∑ k = 1 4 s k ( ∑ ν ( a ν ) k j 2 ) = 0. \sum_{k=1}^{4}s_k\left(\sum_\nu(a_\nu)_{kj}^2\right)=0. k = 1 ∑ 4 s k ( ν ∑ ( a ν ) k j 2 ) = 0. The observation forces all four sums of squares to vanish, hence every entry in column j j j a ν a_\nu a ν j j j B B B B B B 0 0 0
Absence of real primitive ideals and real maximal left ideals. Every primitive ideal of B B B M 4 ( m ) M_4(\mathfrak m) M 4 ( m ) m \mathfrak m m C C C R \mathbb{R} R C \mathbb{C} C i i i 1 ∗ 1 + i ∗ i = 0 1^*1+i^*i=0 1 ∗ 1 + i ∗ i = 0 m \mathfrak m m ( a , b ) ∈ ( R × ) 2 (a,b)\in(\mathbb{R}^{\times})^2 ( a , b ) ∈ ( R × ) 2
⟨ 1 , a , b , − a b ⟩ . \langle1,a,b,-ab\rangle. ⟨ 1 , a , b , − ab ⟩ . This form is always indefinite: positive definiteness would require a , b , − a b > 0 a,b,-ab>0 a , b , − ab > 0 1 1 1 z z z h h h z z z h ≠ 0 h\ne0 h = 0 h ∗ h = 0 h^*h=0 h ∗ h = 0
Positive involutions and *-representations
Let A A A ∗ * ∗ K K K F = { a ∈ K : a ∗ = a } F=\{a\in K:a^*=a\} F = { a ∈ K : a ∗ = a } Trd A / K ( a ∗ ) = Trd A / K ( a ) ∗ \operatorname{Trd}_{A/K}(a^*)=\operatorname{Trd}_{A/K}(a)^* Trd A / K ( a ∗ ) = Trd A / K ( a ) ∗ F F F
T ∗ ( x ) = Trd A / K ( x ∗ x ) ( x ∈ A ) T_*(x)=\operatorname{Trd}_{A/K}(x^*x)\qquad(x\in A) T ∗ ( x ) = Trd A / K ( x ∗ x ) ( x ∈ A ) is a quadratic form over F F F involution trace form . The involution is positive at an ordering of F F F T ∗ T_* T ∗ [32 Definition 1.1] . Positivity implies formal reality: applying Trd A / K \operatorname{Trd}_{A/K} Trd A / K ∑ i x i ∗ x i = 0 \sum_i x_i^*x_i=0 ∑ i x i ∗ x i = 0 x i = 0 x_i=0 x i = 0
Remark 4.13 (Normalizations of the trace form). Our normalization agrees with Procesi–Schacher [32 Definition 1.1] and Klep–Unger [23 Section 2] for involutions of the first kind. In the unitary case they use 2 T ∗ 2T_* 2 T ∗
For k = R k=\mathbb{R} k = R [9 Proposition 2.7] .
Lemma 4.14 (Finite-dimensional real algebras over a real closed field). Let k k k A A A k k k k k k
If A A A M N ( k ) M_N(k) M N ( k )
If A A A A A A k k k k k k
Proof. (1) Suppose A A A A A A e e e e ∗ e = 0 e^*e=0 e ∗ e = 0 A A A K K K k k k k ( i ) k(i) k ( i ) K K K i i i 1 ∗ 1 + i ∗ i = 0 1^*1+i^*i=0 1 ∗ 1 + i ∗ i = 0 k k k 0 ≠ x ∈ A 0 \ne x \in A 0 = x ∈ A s = x ∗ x s=x^*x s = x ∗ x k [ s ] k[s] k [ s ] k k k k ( i ) k(i) k ( i ) k ( i ) k(i) k ( i ) e ≠ 0 e \ne0 e = 0 u u u e 2 = e e^2=e e 2 = e u 2 = − e u^2=-e u 2 = − e e ∗ e + u ∗ u = 0 e^*e+u^*u=0 e ∗ e + u ∗ u = 0 s = ∑ j λ j e j s=\sum_j\lambda_je_j s = ∑ j λ j e j λ j ∈ k \lambda_j\in k λ j ∈ k e j e_j e j λ j < 0 \lambda_j<0 λ j < 0 μ = − λ j ∈ k \mu=\sqrt{-\lambda_j}\in k μ = − λ j ∈ k ( x e j ) ∗ ( x e j ) + ( μ e j ) ∗ ( μ e j ) = 0 (xe_j)^*(xe_j)+(\mu e_j)^*(\mu e_j)=0 ( x e j ) ∗ ( x e j ) + ( μ e j ) ∗ ( μ e j ) = 0 λ j ≥ 0 \lambda_j\ge0 λ j ≥ 0 s ≠ 0 s\ne0 s = 0 Trd A / K ( e j ) \operatorname{Trd}_{A/K}(e_j) Trd A / K ( e j )
T ∗ ( x ) = ∑ j λ j Trd A / K ( e j ) > 0. T_*(x)=\sum_j\lambda_j\operatorname{Trd}_{A/K}(e_j)>0. T ∗ ( x ) = j ∑ λ j Trd A / K ( e j ) > 0. The polarization ⟨ x , y ⟩ = 1 2 Trd A / K ( x ∗ y + y ∗ x ) \langle x,y\rangle=\frac{1}{2}\operatorname{Trd}_{A/K}(x^*y+y^*x) ⟨ x , y ⟩ = 2 1 Trd A / K ( x ∗ y + y ∗ x ) k k k ⟨ a x , y ⟩ = ⟨ x , a ∗ y ⟩ \langle ax,y\rangle=\langle x,a^*y\rangle ⟨ a x , y ⟩ = ⟨ x , a ∗ y ⟩ k k k ∗ * ∗ A ↪ M dim k A ( k ) A\hookrightarrow M_{\dim_k A}(k) A ↪ M d i m k A ( k )
(2) For simple real A A A k k k ∑ i x i ∗ x i = 0 \sum_i x_i^*x_i=0 ∑ i x i ∗ x i = 0 x i = 0 x_i=0 x i = 0
Let R R R k k k k k k E E E k k k ∗ * ∗ R R R E E E k k k π : R → End E ( V ) \pi:R\to\operatorname{End}_E(V) π : R → End E ( V ) V V V E E E
⟨ π ( a ) u , v ⟩ = ⟨ u , π ( a ∗ ) v ⟩ ( a ∈ R ) . \langle\pi(a)u,v\rangle=\langle u,\pi(a^*)v\rangle\qquad(a\in R). ⟨ π ( a ) u , v ⟩ = ⟨ u , π ( a ∗ ) v ⟩ ( a ∈ R ) . It is irreducible if V V V E E E E E E π \pi π ∗ * ∗ R → ( M n ( E ) , T ) R\to(M_n(E),{}^{\mathsf T}) R → ( M n ( E ) , T ) E E E ∗ * ∗ E n E^n E n E E E
Lemma 4.15 (Orthogonal decomposition). Let π \pi π ∗ * ∗ R R R E E E k k k
(1) π \pi π ∗ * ∗ ker π \ker\pi ker π
(2) If π \pi π ker π \ker\pi ker π E π ( R ) E\pi(R) E π ( R ) R R R E E E k k k ker π \ker\pi ker π k k k
The decomposition in (1) also holds over C \mathbb{C} C
Proof. (1) If W ⊆ V W\subseteq V W ⊆ V W ⊥ W^\perp W ⊥ V = W ⊕ W ⊥ V=W\oplus W^\perp V = W ⊕ W ⊥ dim E V \dim_E V dim E V ∑ i a i ∗ a i ∈ ker π \sum_i a_i^*a_i\in\ker\pi ∑ i a i ∗ a i ∈ ker π ∑ i ⟨ π ( a i ) v , π ( a i ) v ⟩ = 0 \sum_i\langle\pi(a_i)v,\pi(a_i)v\rangle=0 ∑ i ⟨ π ( a i ) v , π ( a i ) v ⟩ = 0 v ∈ V v\in V v ∈ V π ( a i ) = 0 \pi(a_i)=0 π ( a i ) = 0 ker π \ker\pi ker π
(2) If two-sided ideals I , J I,J I , J I J ⊆ ker π IJ\subseteq\ker\pi I J ⊆ ker π J ⊈ ker π J\nsubseteq\ker\pi J ⊈ ker π π ( J ) V \pi(J)V π ( J ) V V V V π ( I ) V = π ( I J ) V = 0 \pi(I)V=\pi(IJ)V=0 π ( I ) V = π ( I J ) V = 0 ker π \ker\pi ker π E π ( R ) E\pi(R) E π ( R ) V V V π ( R ) \pi(R) π ( R ) E E E V V V E π ( R ) E\pi(R) E π ( R ) E π ( R ) ⊆ End E ( V ) E\pi(R)\subseteq\operatorname{End}_E(V) E π ( R ) ⊆ End E ( V ) E π ( R ) E\pi(R) E π ( R ) R R R E E E k k k V V V k k k π ( R ) ≅ R / ker π \pi(R)\cong R/\ker\pi π ( R ) ≅ R / ker π k k k ker π \ker\pi ker π
Lemma 4.16 will be used in Corollaries 4.18–4.19 and Theorem 4.20.
Lemma 4.16 (Specialization). Let k k k R R R k k k k k k φ \varphi φ ∗ * ∗ R R R E E E k k k b ∈ R b\in R b ∈ R φ ( b ) ≠ 0 \varphi(b)\ne0 φ ( b ) = 0 K K K k k k ∗ * ∗ ψ \psi ψ R R R K K K ker φ ⊆ ker ψ \ker\varphi\subseteq\ker\psi ker φ ⊆ ker ψ ψ ( b ) ≠ 0 \psi(b)\ne0 ψ ( b ) = 0
Proof. Extending scalars to a real closure Ω \Omega Ω E E E φ ( R ) ⊆ ( M n ( Ω ) , T ) \varphi(R) \subseteq(M_n(\Omega),{}^T) φ ( R ) ⊆ ( M n ( Ω ) , T ) K K K Ω \Omega Ω k k k k k k k k k R R R C ⊆ Ω C \subseteq\Omega C ⊆ Ω k k k φ ( R ) ⊆ M n ( C ) \varphi(R) \subseteq M_n(C) φ ( R ) ⊆ M n ( C ) φ ( b ) \varphi(b) φ ( b ) c ∈ C c \in C c ∈ C C = k [ t 1 , … , t m ] / ( f 1 , … , f r ) C=k[t_1,\ldots,t_m]/(f_1,\ldots,f_r) C = k [ t 1 , … , t m ] / ( f 1 , … , f r ) c c c g ( t ) g(t) g ( t ) f 1 = ⋯ = f r = 0 f_1=\cdots=f_r=0 f 1 = ⋯ = f r = 0 g ( t ) z = 1 g(t)z=1 g ( t ) z = 1 Ω \Omega Ω [1 Chapter 5] , one in K K K k k k ε : C → K \varepsilon:C\to K ε : C → K ε ( c ) ≠ 0 \varepsilon(c)\ne0 ε ( c ) = 0 ε \varepsilon ε ψ = ε ∘ φ \psi=\varepsilon\circ\varphi ψ = ε ∘ φ ∗ \ast ∗ K K K ker φ ⊆ ker ψ \ker\varphi\subseteq\ker\psi ker φ ⊆ ker ψ ψ ( b ) ≠ 0 \psi(b)\ne0 ψ ( b ) = 0
Positive real primes and the real Jacobson property
We now ask when real maximal left ideals exist and when R R R k k k R R R k k k k k k B B B central simple ring of quotients is ( Z ( B ) ∖ 0 ) − 1 B (Z(B)\setminus0)^{-1}B ( Z ( B ) ∖ 0 ) − 1 B [31 ] ; an involution of B B B T T T R R R positive if the involution induced on the central simple ring of quotients of R / T R/T R / T
We first describe positive real primes by ∗ \ast ∗ Theorem 4.20 ).
Proposition 4.17 (Positive real primes and representations). For a real prime two-sided ideal T T T R R R
T T T
T T T ∗ \ast ∗ k k k
T T T ∗ \ast ∗ k k k
In (2) and (3), the ordered field can be taken to be the fixed centre of the central simple ring of quotients of R / T R/T R / T
Proof. Put B = R / T B=R/T B = R / T Q Q Q F F F Q Q Q k k k k ⊆ F k\subseteq F k ⊆ F
(1)⇒ \Rightarrow ⇒ F F F T ∗ T_\ast T ∗ Q Q Q
⟨ x , y ⟩ = 1 2 Trd Q / Z ( Q ) ( x ∗ y + y ∗ x ) \langle x,y\rangle=\frac{1}{2}\operatorname{Trd}_{Q/Z(Q)}(x^\ast y+y^\ast x) ⟨ x , y ⟩ = 2 1 Trd Q / Z ( Q ) ( x ∗ y + y ∗ x ) is a positive definite symmetric F F F Q Q Q ⟨ a x , y ⟩ = ⟨ x , a ∗ y ⟩ \langle ax,y\rangle=\langle x,a^\ast y\rangle ⟨ a x , y ⟩ = ⟨ x , a ∗ y ⟩ a , x , y ∈ Q a,x,y\in Q a , x , y ∈ Q F B = Q FB=Q F B = Q s ∈ Z ( B ) s\in Z(B) s ∈ Z ( B ) s − 1 = s ∗ ( s ∗ s ) − 1 s^{-1}=s^\ast(s^\ast s)^{-1} s − 1 = s ∗ ( s ∗ s ) − 1 s ∗ s ∈ F s^\ast s\in F s ∗ s ∈ F L L L Q Q Q F F F L L L B B B F B = Q FB=Q F B = Q R → B → End F ( L ) R\to B\to\operatorname{End}_F(L) R → B → End F ( L ) ∗ \ast ∗ F F F T T T Q Q Q L L L
(3)⇒ \Rightarrow ⇒
(2)⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ φ \varphi φ ∗ \ast ∗ T T T Ω \Omega Ω T T T φ \varphi φ T T T ψ \psi ψ T T T B B B ψ ( R ) \psi(R) ψ ( R ) A = Ω B A=\Omega B A = Ω B ∗ \ast ∗ ( M n ( Ω ) , T ) (M_n(\Omega),{}^T) ( M n ( Ω ) , T ) Z ( B ) Z(B) Z ( B ) Ω B = A \Omega B=A Ω B = A Z ( B ) Z(B) Z ( B ) Z ( A ) Z(A) Z ( A ) s ∈ Z ( B ) s\in Z(B) s ∈ Z ( B ) A A A B ↪ A B\hookrightarrow A B ↪ A Q = ( Z ( B ) ∖ { 0 } ) − 1 B ↪ A Q=(Z(B)\setminus\{0\})^{-1}B\hookrightarrow A Q = ( Z ( B ) ∖ { 0 } ) − 1 B ↪ A b / s ↦ b s − 1 b/s\mapsto bs^{-1} b / s ↦ b s − 1 b s − 1 = 0 bs^{-1}=0 b s − 1 = 0 b = 0 b=0 b = 0 A A A Ω \Omega Ω B B B Ω ⊆ Z ( A ) \Omega\subseteq Z(A) Ω ⊆ Z ( A )
Q ⊗ Z ( Q ) Z ( A ) ⟶ A . Q\otimes_{Z(Q)}Z(A)\longrightarrow A. Q ⊗ Z ( Q ) Z ( A ) ⟶ A . Its domain is central simple over Z ( A ) Z(A) Z ( A ) Q Q Q A A A A A A Ω \Omega Ω a ≠ 0 a \ne0 a = 0 F ⊆ Ω F \subseteq\Omega F ⊆ Ω Trd Q / Z ( Q ) ( q ∗ q ) > 0 \operatorname{Trd}_{Q/Z(Q)}(q^{*}q) > 0 Trd Q / Z ( Q ) ( q ∗ q ) > 0 q ≠ 0 q \ne0 q = 0 Q Q Q
For primitive ideals, maximality allows specialization to a real closure without changing the kernel.
Corollary 4.18 (Positive real primitive ideals). The positive real primitive ideals of R R R ∗ * ∗ k k k k k k
Proof. Let J J J J J J ∗ * ∗ b = 1 b = 1 b = 1 ∗ * ∗ ψ \psi ψ k k k J ⊆ ker ψ J \subseteq\ker\psi J ⊆ ker ψ ψ \psi ψ J J J J J J ∗ * ∗ k k k k k k R / J R/J R / J ∗ * ∗ R / J R/J R / J k k k J J J
Corollary 4.19 (Existence). R R R ∗ * ∗ k k k k k k R R R In this case R R R k k k
Proof. Lemma 4.16 with b = 1 b = 1 b = 1 ∗ * ∗ k k k
We now pass from individual kernels to intersections. The main step is to combine a family of representations whose kernels have prime intersection into a single representation with the same kernel.
Theorem 4.20 (Positive real primes and the real Jacobson property). Let k k k R R R k k k k k k
(i) A real prime two-sided ideal T T T R R R
(ii) Every real prime two-sided ideal of R R R R R R R R R
Proof. (i) Suppose T T T ∗ * ∗ T T T b ∉ T b \notin T b ∈ / T ∗ * ∗ k k k T T T b b b b b b T T T
Conversely, by Corollary 4.18, write T T T ∗ * ∗ k k k R R R d d d ∗ * ∗ π \pi π L L L k k k A = L π ( R ) A = L\pi(R) A = L π ( R ) Z ( A ) Z(A) Z ( A ) L L L L ( i ) L(i) L ( i ) dim L A ≤ 2 ( d / 2 ) 2 ≤ d 2 \dim_L A \le2(d/2)^2 \le d^2 dim L A ≤ 2 ( d /2 ) 2 ≤ d 2 [13 Part B, Lemma 4.14(a),(c)] . The representation space is cyclic over A A A d 2 d^2 d 2
Choose a common multiple N N N 1 , … , d 2 1,\ldots,d^2 1 , … , d 2 N N N φ : R → ( M N ( L φ ) , T ) \varphi\colon R \to(M_N(L_\varphi), \mathsf{T}) φ : R → ( M N ( L φ ) , T ) C ⊆ ∏ φ L φ C \subseteq\prod_\varphi L_\varphi C ⊆ ∏ φ L φ k k k R R R R / T ↪ M N ( C ) R/T \hookrightarrow M_N(C) R / T ↪ M N ( C ) C C C S S S C C C S − 1 C S^{-1}C S − 1 C E 1 , … , E s E_1,\ldots,E_s E 1 , … , E s [36 Tags 00EU, 00EW and 02LX] , and C ↪ S − 1 C C \hookrightarrow S^{-1}C C ↪ S − 1 C ∑ j ( b j / u ) 2 = 0 \sum_j (b_j/u)^2=0 ∑ j ( b j / u ) 2 = 0 u ∈ S u\in S u ∈ S ∑ j b j 2 = 0 \sum_j b_j^2=0 ∑ j b j 2 = 0 C C C b j = 0 b_j=0 b j = 0 E i E_i E i
Choose an ordering of each E i E_i E i R → M N ( C ) → M N ( E i ) R\to M_N(C)\to M_N(E_i) R → M N ( C ) → M N ( E i ) ∗ * ∗ T T T C ↪ ∏ i E i C\hookrightarrow\prod_i E_i C ↪ ∏ i E i T T T T T T T T T
(ii) Real primitive ideals are real prime. If every real prime ideal is positive, part (i) writes each of them as an intersection of real primitive ideals, so R R R R R R
Positivity is hard to verify directly; for Azumaya algebras, a condition on reduced traces of hermitian squares suffices.
Corollary 4.21 (Procesi–Schacher condition). Let R R R C C C k k k ∗ * ∗ k k k R R R Trd R / C ( a ∗ a ) 1 R ∈ Σ R ∗ R \operatorname{Trd}_{R/C}(a^*a)1_R\in\Sigma R^*R Trd R / C ( a ∗ a ) 1 R ∈ Σ R ∗ R a ∈ R a\in R a ∈ R R R R R R R Theorem 4.20 (ii).
Proof. Since R R R C C C k k k T T T p = T ∩ C p=T\cap C p = T ∩ C T = p R T=pR T = pR R / T R/T R / T Q = K ⊗ C / p R / T Q=K\otimes_{C/p}R/T Q = K ⊗ C / p R / T K = Frac ( C / p ) K=\operatorname{Frac}(C/p) K = Frac ( C / p ) F F F Q Q Q a / s a/s a / s 0 ≠ s ∈ C / p 0\ne s\in C/p 0 = s ∈ C / p Q Q Q Trd ( ( a / s ) ∗ ( a / s ) ) = Trd ( a ∗ a ) / ( s ∗ s ) \operatorname{Trd}((a/s)^*(a/s))=\operatorname{Trd}(a^*a)/(s^*s) Trd (( a / s ) ∗ ( a / s )) = Trd ( a ∗ a ) / ( s ∗ s ) Q Q Q s s s P = F ∩ Σ Q ∗ Q P=F\cap\Sigma Q^*Q P = F ∩ Σ Q ∗ Q Trd ( q ∗ q ) \operatorname{Trd}(q^*q) Trd ( q ∗ q ) F F F − 1 -1 − 1 α = ∑ i a i ∗ a i \alpha=\sum_i a_i^*a_i α = ∑ i a i ∗ a i β = ∑ j b j ∗ b j \beta=\sum_j b_j^*b_j β = ∑ j b j ∗ b j P P P α β = ∑ j b j ∗ α b j = ∑ i , j ( a i b j ) ∗ ( a i b j ) ∈ P \alpha\beta=\sum_j b_j^*\alpha b_j=\sum_{i,j}(a_ib_j)^*(a_ib_j)\in P α β = ∑ j b j ∗ α b j = ∑ i , j ( a i b j ) ∗ ( a i b j ) ∈ P P P P T T T
Remark 4.22 (Trace squares, positivity and real Jacobsonness). By Corollary 4.21 and Theorem 4.20 (ii), the trace-square condition implies positivity of every real prime ideal, which in turn implies real Jacobsonness. Neither implication can be reversed, even for central simple algebras.
For central simple algebras, the trace-square condition is equivalent to the Procesi–Schacher conjecture [23 Remark 2.5] . Klep and Unger give a split degree-three counterexample whose involution is positive at some ordering but whose trace-square condition fails [23 Theorem 3.2] . Since zero is its only proper prime ideal, this shows that the first implication cannot be reversed.
For the second implication, let k = R ( x , y ) k=\mathbb{R}(x,y) k = R ( x , y ) A = M 4 ( k ) A=M_4(k) A = M 4 ( k ) ⟨ 1 , x , y , − x y ⟩ \langle1,x,y,-xy\rangle ⟨ 1 , x , y , − x y ⟩ 0 0 0 x , y , − x y x,y,-xy x , y , − x y E 21 , E 31 , E 41 E_{21},E_{31},E_{41} E 21 , E 31 , E 41 k k k ∗ * ∗ k k k ⟨ 1 , x , y , − x y ⟩ \langle1,x,y,-xy\rangle ⟨ 1 , x , y , − x y ⟩ [24 p. 68] .
Free associative algebras
This section proves Theorem D, except for the algorithmic assertion in part (3), proved in Appendix A. Lewin’s theorem splits off the finite-dimensional semisimple part. For at least two generators, irreducible completions of partial operators then give separating finite-dimensional simple quotients.
Fix F = k ⟨ x 1 , … , x g ⟩ F = k\langle x_1,\ldots,x_g\rangle F = k ⟨ x 1 , … , x g ⟩ g < ∞ g < \infty g < ∞ k k k F n F^n F n F 1 × n F^{1\times n} F 1 × n Table 1 summarizes the results of this section and Section 6 for finitely generated submodules of F n F^n F n n = 1 n=1 n = 1 rad N \operatorname{rad} N rad N N r e \sqrt[\mathrm{re}]{N} re N
Table 1. Separation, evaluations, and radicals over free associative algebras.
All assertions in the table concern finitely generated submodules; they do not assert a property of all submodules of the free module. In the real column, k = R k = \mathbb{R} k = R C \mathbb{C} C ∗ * ∗ No involution column of Table 1 are new. The extension of real prime separation to arbitrary fields and involutions is also new. For the real Nullstellensatz and real-radical bounds, we give direct proofs of results of [8 , 19 ] .
Remark 5.1 (Reduction to free algebras). Question 1 reduces to free associative rings: every unital ring is a quotient of some Z ⟨ X ⟩ \mathbb{Z}\langle X\rangle Z ⟨ X ⟩ k k k k k k k k k k k k
A finitely generated left ideal in a finitely presented quotient algebra need not have a finitely generated inverse image in the presenting free algebra. Indeed, if R = F / J R=F/J R = F / J F F F I ⊆ R I\subseteq R I ⊆ R f 1 , … , f r f_1,\ldots,f_r f 1 , … , f r J + ∑ i F f i J+\sum_i Ff_i J + ∑ i F f i R R R J J J F = k ⟨ x , y ⟩ F=k\langle x,y\rangle F = k ⟨ x , y ⟩ J = F x F J=FxF J = F x F ⨁ m ≥ 0 F x y m \bigoplus_{m\ge0}Fxy^m ⨁ m ≥ 0 F x y m F / J ≅ k [ y ] F/J \cong k[y] F / J ≅ k [ y ] R R R F F F
Prime separation
Theorem 5.2 answers Question 1 affirmatively for finitely generated semiprime left ideals of F F F
Theorem 5.2 (Free algebras). Every finitely presented semiprime left F F F Consequently, if N ⊆ M N \subseteq M N ⊆ M M / N M/N M / N N N N M M M
The main technical ingredient of the proof is Lemma 5.3, which uses Lewin’s theorem: every finitely presented F F F [25 Theorem 2] . Note that this free submodule need not be a direct summand.
Lemma 5.3 (Finite-dimensional part). Let F = k ⟨ x 1 , … , x g ⟩ F = k\langle x_1,\ldots,x_g\rangle F = k ⟨ x 1 , … , x g ⟩ g ≥ 1 g \ge1 g ≥ 1 F F F Q Q Q Q = D ⊕ K Q = D \oplus K Q = D ⊕ K D D D K K K
Proof. By [25 Theorem 2] , choose a free submodule U ⊆ Q U \subseteq Q U ⊆ Q Q / U Q/U Q / U D D D Q Q Q F F F D ∩ U D \cap U D ∩ U E ∩ U E \cap U E ∩ U E E E E ∩ U E \cap U E ∩ U U U U D ∩ U = 0 D \cap U = 0 D ∩ U = 0 D D D Q / U Q/U Q / U
If D = 0 D = 0 D = 0 K = Q K = Q K = Q T = Ann F ( D ) T = \operatorname{Ann}_F(D) T = Ann F ( D ) T T T F → End k ( D ) F \to\operatorname{End}_k(D) F → End k ( D ) T T T F / T F/T F / T End k ( D ) \operatorname{End}_k(D) End k ( D ) F / T F/T F / T Q Q Q D ∩ T Q = 0 D \cap TQ = 0 D ∩ T Q = 0 F / T F/T F / T
0 ⟶ D ⟶ Q / T Q ⟶ Q / ( D + T Q ) ⟶ 0 0 \longrightarrow D \longrightarrow Q/TQ \longrightarrow Q/(D+TQ) \longrightarrow0 0 ⟶ D ⟶ Q / T Q ⟶ Q / ( D + T Q ) ⟶ 0 is exact. Since F / T F/T F / T Q / T Q → D Q/TQ \to D Q / T Q → D r : Q → D r: Q \to D r : Q → D Q = D ⊕ K Q = D \oplus K Q = D ⊕ K K = ker r K = \ker r K = ker r D D D F / T F/T F / T K K K Q Q Q D D D D ∩ K = 0 D \cap K = 0 D ∩ K = 0
For 0 ≠ L ⊆ K 0 \ne L \subseteq K 0 = L ⊆ K L ∩ U = 0 L \cap U = 0 L ∩ U = 0 L L L Q / U Q/U Q / U L L L D ∩ K = 0 D \cap K = 0 D ∩ K = 0 L ∩ U ≠ 0 L \cap U \ne0 L ∩ U = 0 K ∩ U K \cap U K ∩ U K K K K ≠ 0 K \ne0 K = 0 K ∩ U K \cap U K ∩ U F F F F F F U U U K K K □ \square □
Proof of Theorem 5.2. For g ≥ 1 g \ge1 g ≥ 1 K K K M / N = ⨁ i = 1 r H i M/N = \bigoplus_{i=1}^{r} H_i M / N = ⨁ i = 1 r H i P i P_i P i M → M / N → H i M \to M/N \to H_i M → M / N → H i M / P i ≅ H i M/P_i \cong H_i M / P i ≅ H i P i P_i P i ⋂ i = 1 r P i = N \bigcap_{i=1}^{r} P_i = N ⋂ i = 1 r P i = N N N N g = 0 g = 0 g = 0 F = k F = k F = k □ \square □
Example 5.4 (Finite free modules). Every finitely generated semiprime submodule N ⊆ F n N \subseteq F^n N ⊆ F n F n / N F^n/N F n / N Q = F n / N = ⨁ i H i Q = F^n/N = \bigoplus_i H_i Q = F n / N = ⨁ i H i C i = ⨁ j ≠ i H j C_i = \bigoplus_{j \ne i} H_j C i = ⨁ j = i H j P i P_i P i C i C_i C i F n F^n F n N N N C i C_i C i
projecting a finite generating set of Q Q Q C i C_i C i C i C_i C i b i 1 , … , b i t i ∈ F n b_{i1},\ldots,b_{it_i}\in F^n b i 1 , … , b i t i ∈ F n P i = N + ∑ j F b i j P_i=N+\sum_j Fb_{ij} P i = N + ∑ j F b ij P i P_i P i n = 1 n=1 n = 1 M M M N ⊆ M N\subseteq M N ⊆ M N N N M M M M / N M/N M / N
Maximal submodules and representations
To describe points and evaluation, let π : F → End k ( V π ) \pi:F\to\operatorname{End}_k(V_\pi) π : F → End k ( V π ) k k k 0 < dim k V π < ∞ 0<\dim_k V_\pi<\infty 0 < dim k V π < ∞ v = ( v 1 , … , v n ) ∈ ( V π ) n v=(v_1,\ldots,v_n)\in(V_\pi)^n v = ( v 1 , … , v n ) ∈ ( V π ) n ε π , v : F n → V π \varepsilon_{\pi,v}:F^n\to V_\pi ε π , v : F n → V π
ε π , v ( f 1 , … , f n ) = ∑ j = 1 n π ( f j ) v j , L π , v = ker ε π , v . (3) \varepsilon_{\pi,v}(f_1,\ldots,f_n)=\sum_{j=1}^{n}\pi(f_j)v_j,\qquad L_{\pi,v}=\ker\varepsilon_{\pi,v}. \tag*{(3)} ε π , v ( f 1 , … , f n ) = j = 1 ∑ n π ( f j ) v j , L π , v = ker ε π , v . (3) A point is a pair ( π , v ) (\pi,v) ( π , v ) v ≠ 0 v\ne0 v = 0 irreducible if V π V_\pi V π real when π \pi π ∗ * ∗ zero of N N N ε π , v ( N ) = 0 \varepsilon_{\pi,v}(N)=0 ε π , v ( N ) = 0 p ∈ F n p\in F^n p ∈ F n ε π , v ( p ) = 0 \varepsilon_{\pi,v}(p)=0 ε π , v ( p ) = 0
We write V V V V π V_\pi V π π \pi π π \pi π X i = π ( x i ) X_i=\pi(x_i) X i = π ( x i ) π ( f ) = f ( X ) \pi(f)=f(X) π ( f ) = f ( X ) k k k
Proposition 5.5 (Maximal kernels). For a maximal submodule L ⊂ F n L\subset F^n L ⊂ F n
L L L
L L L k k k
L = L π , v L=L_{\pi,v} L = L π , v π \pi π v ≠ 0 v\ne0 v = 0
These submodules are classified by such pairs ( π , v ) (\pi,v) ( π , v ) L π , v = L π ′ , v ′ L_{\pi,v}=L_{\pi',v'} L π , v = L π ′ , v ′ T T T T π ( f ) = π ′ ( f ) T T\pi(f)=\pi'(f)T T π ( f ) = π ′ ( f ) T f ∈ F f\in F f ∈ F T v j = v j ′ Tv_j=v'_j T v j = v j ′ j j j
Proof. (1)⇒ \Rightarrow ⇒ L L L V = F n / L V=F^n/L V = F n / L g ≥ 1 g\ge1 g ≥ 1 V V V F F F F F F F x 1 Fx_1 F x 1 V V V g = 0 g=0 g = 0
(2)⇒ \Rightarrow ⇒ F F F V = F n / L V=F^n/L V = F n / L π \pi π v j = e j + L v_j=e_j+L v j = e j + L v v v ε π , v ( f ) = f + L \varepsilon_{\pi,v}(f)=f+L ε π , v ( f ) = f + L L = L π , v L=L_{\pi,v} L = L π , v
(3)⇒ \Rightarrow ⇒ v ≠ 0 v\ne0 v = 0 ε π , v \varepsilon_{\pi,v} ε π , v V V V L L L
(2)⇒ \Rightarrow ⇒ s 1 , … , s t s_1,\ldots,s_t s 1 , … , s t b 1 , … , b t ∈ F n b_1,\ldots,b_t\in F^n b 1 , … , b t ∈ F n e j + L = ∑ ℓ c j ℓ s ℓ e_j+L=\sum_\ell c_{j\ell}s_\ell e j + L = ∑ ℓ c j ℓ s ℓ x i b ℓ = ∑ r a i ℓ r b r x_i b_\ell=\sum_r a_{i\ell r}b_r x i b ℓ = ∑ r a i ℓ r b r
e j − ∑ ℓ c j ℓ b ℓ , x i b ℓ − ∑ r a i ℓ r b r e_j-\sum_\ell c_{j\ell}b_\ell,\qquad x_i b_\ell-\sum_r a_{i\ell r}b_r e j − ℓ ∑ c j ℓ b ℓ , x i b ℓ − r ∑ a i ℓ r b r generate L L L F F F b ℓ b_\ell b ℓ t t t t t t F n / L F^n/L F n / L F n F^n F n □ \square □
For fixed π \pi π Aut F ( V ) \operatorname{Aut}_F(V) Aut F ( V ) k k k N N N p ( 1 ) , … , p ( r ) p^{(1)},\ldots,p^{(r)} p ( 1 ) , … , p ( r ) N ⊆ L π , v N\subseteq L_{\pi,v} N ⊆ L π , v ∑ j π ( p j ( ℓ ) ) v j = 0 \sum_j\pi(p_j^{(\ell)})v_j=0 ∑ j π ( p j ( ℓ ) ) v j = 0 ℓ \ell ℓ
The Jacobson property
Separation by finite-dimensional simple modules rests on two lemmas. The first reduces it to extending finitely many partial operators; it is used again, with a different extension, in the real case.
For Q = F n / N Q=F^n/N Q = F n / N m ≥ 0 m\ge0 m ≥ 0 Q ≤ m Q_{\le m} Q ≤ m m m m x i Q ≤ m − 1 ⊆ Q ≤ m x_iQ_{\le m-1}\subseteq Q_{\le m} x i Q ≤ m − 1 ⊆ Q ≤ m
Lemma 5.6 (Truncation). Let N ⊆ F n N\subseteq F^n N ⊆ F n e ≥ 1 e\ge1 e ≥ 1 Q = F n / N Q=F^n/N Q = F n / N H H H k k k Q ≤ e Q_{\le e} Q ≤ e k k k X 1 , … , X g X_1,\ldots,X_g X 1 , … , X g X i u = x i u X_i u=x_i u X i u = x i u u ∈ Q ≤ e − 1 u\in Q_{\le e-1} u ∈ Q ≤ e − 1 H H H F F F X i X_i X i F F F Q → H Q\to H Q → H Q ≤ e Q_{\le e} Q ≤ e
Proof. Define ϕ ~ : F n → H \widetilde{\phi}:F^n\to H ϕ : F n → H ϕ ~ ( f 1 , … , f n ) = ∑ j f j ( X ) ( e j + N ) \widetilde{\phi}(f_1,\ldots,f_n)=\sum_j f_j(X)(e_j+N) ϕ ( f 1 , … , f n ) = ∑ j f j ( X ) ( e j + N ) w w w ∣ w ∣ ≤ e |w|\le e ∣ w ∣ ≤ e ∣ w ∣ |w| ∣ w ∣ ϕ ~ ( w e j ) = w e j + N \widetilde{\phi}(we_j)=we_j+N ϕ ( w e j ) = w e j + N w = x i w ′ w=x_iw' w = x i w ′ w ′ e j + N ∈ Q ≤ e − 1 w'e_j+N\in Q_{\le e-1} w ′ e j + N ∈ Q ≤ e − 1 ϕ ~ ( w e j ) = X i ( w ′ e j + N ) = x i w ′ e j + N \widetilde{\phi}(we_j)=X_i(w'e_j+N)=x_iw'e_j+N ϕ ( w e j ) = X i ( w ′ e j + N ) = x i w ′ e j + N ϕ ~ \widetilde{\phi} ϕ e e e N N N N N N
The second lemma supplies irreducible extensions. It requires no involution.
Lemma 5.7 (Irreducible completion). Let g ≥ 2 g\ge2 g ≥ 2 V ⊆ W V\subseteq W V ⊆ W k k k T i : V → W T_i:V\to W T i : V → W L ⊆ V L\subseteq V L ⊆ V T i L ⊆ L T_iL\subseteq L T i L ⊆ L i i i h ≥ max { 1 , 2 dim k W } h\ge\max\{1,2\dim_k W\} h ≥ max { 1 , 2 dim k W } h h h k k k W W W
Proof. Extend the T i T_i T i B i ∈ End k ( W ) B_i\in\operatorname{End}_k(W) B i ∈ End k ( W ) w = dim k W w=\dim_k W w = dim k W m = h − w ≥ 1 m=h-w\ge1 m = h − w ≥ 1 r = m − w ≥ 0 r=m-w\ge0 r = m − w ≥ 0 χ ( t ) \chi(t) χ ( t ) B 1 B_1 B 1
f ( t ) = 1 + t r χ ( t ) , U = k [ t ] / ( f ) . f(t)=1+t^r\chi(t),\qquad U=k[t]/(f). f ( t ) = 1 + t r χ ( t ) , U = k [ t ] / ( f ) . The polynomial f f f m m m U U U 1 , t , … , t m − 1 1,t,\ldots,t^{m-1} 1 , t , … , t m − 1 U U U A A A t t t
J ( ∑ j = 0 m − 1 a j t j ) = a m − 1 1. J\left(\sum_{j=0}^{m-1}a_jt^j\right)=a_{m-1}1. J ( j = 0 ∑ m − 1 a j t j ) = a m − 1 1. The pair ( A , J ) (A,J) ( A , J ) u u u d < m d<m d < m t m − 1 − d t^{m-1-d} t m − 1 − d m − 1 m-1 m − 1 J J J 1 1 1 A A A U U U
Since m ≥ w m\ge w m ≥ w
R : U ↠ W , C : W → U , ker C = V . R:U\twoheadrightarrow W,\qquad C:W\to U,\qquad\ker C=V. R : U ↠ W , C : W → U , ker C = V . For C C C W → W / V W\to W/V W → W / V W / V ↪ U W/V\hookrightarrow U W / V ↪ U H = W ⊕ U H=W\oplus U H = W ⊕ U
X 1 = ( B 1 0 0 A ) , X 2 = ( B 2 R C J ) , X i = B i ⊕ 0 ( i ≥ 3 ) . X_1=\begin{pmatrix}B_1&0\\0&A\end{pmatrix},\qquad
X_2=\begin{pmatrix}B_2&R\\C&J\end{pmatrix},\qquad
X_i=B_i\oplus0\quad(i\ge3). X 1 = ( B 1 0 0 A ) , X 2 = ( B 2 C R J ) , X i = B i ⊕ 0 ( i ≥ 3 ) . These operators extend the T i T_i T i C C C V V V P : H → U P:H\to U P : H → U W W W
− X 1 r χ ( X 1 ) = ( − B 1 r χ ( B 1 ) 0 0 − A r χ ( A ) ) = ( 0 0 0 I U ) = P . -X_1^r\chi(X_1)=
\begin{pmatrix}
-B_1^r\chi(B_1)&0\\
0&-A^r\chi(A)
\end{pmatrix}
=
\begin{pmatrix}
0&0\\
0&I_U
\end{pmatrix}
=P. − X 1 r χ ( X 1 ) = ( − B 1 r χ ( B 1 ) 0 0 − A r χ ( A ) ) = ( 0 0 0 I U ) = P . Here χ ( B 1 ) = 0 \chi(B_1)=0 χ ( B 1 ) = 0 f ( A ) = I U + A r χ ( A ) = 0 f(A)=I_U+A^r\chi(A)=0 f ( A ) = I U + A r χ ( A ) = 0 L ⊆ H L\subseteq H L ⊆ H X i X_i X i P P P X 1 X_1 X 1 P P P 1 − P 1-P 1 − P L L L
L = ( L ∩ W ) ⊕ ( L ∩ U ) . L=(L\cap W)\oplus(L\cap U). L = ( L ∩ W ) ⊕ ( L ∩ U ) . The subspace L ∩ U L \cap U L ∩ U A = X 1 ∣ U A = X_1|_U A = X 1 ∣ U J = P X 2 ∣ U J = PX_2|_U J = P X 2 ∣ U U U U L ⊆ W L \subseteq W L ⊆ W x ∈ L x \in L x ∈ L P X 2 x = C x PX_2x = Cx P X 2 x = C x X 2 x ∈ L ⊆ W X_2x \in L \subseteq W X 2 x ∈ L ⊆ W P X 2 x = 0 PX_2x = 0 P X 2 x = 0 L ⊆ ker C = V L \subseteq\ker C = V L ⊆ ker C = V X i ∣ V = T i X_i|_V = T_i X i ∣ V = T i L L L T i T_i T i L = 0 L = 0 L = 0 U ⊆ L U \subseteq L U ⊆ L u ∈ U u \in U u ∈ U X 2 u = R u + J u X_2u = Ru + Ju X 2 u = R u + J u ( 1 − P ) X 2 u = R u (1-P)X_2u = Ru ( 1 − P ) X 2 u = R u 1 − P 1-P 1 − P W W W X 2 X_2 X 2 1 − P 1-P 1 − P L L L
W = R ( U ) = ( 1 − P ) X 2 ( U ) ⊆ L , W = R(U) = (1-P)X_2(U) \subseteq L, W = R ( U ) = ( 1 − P ) X 2 ( U ) ⊆ L , so L = H L = H L = H k k k
Theorem 5.8 (Finite-dimensional simple separation). Every finitely presented semiprime F F F For a finitely generated submodule N ⊆ F n N \subseteq F^n N ⊆ F n
N N N
N N N
N N N
N N N
Proof. For g ≥ 2 g \geq2 g ≥ 2 Q = D ⊕ K Q = D \oplus K Q = D ⊕ K D D D K K K K = F r / P K = F^r/P K = F r / P P P P 0 ≠ q ∈ K 0 \ne q \in K 0 = q ∈ K
Choose e ≥ 1 e \geq1 e ≥ 1 q q q P P P T i : K ≤ e − 1 → K ≤ e T_i: K_{\leq e-1} \to K_{\leq e} T i : K ≤ e − 1 → K ≤ e K ≤ e − 1 K_{\leq e-1} K ≤ e − 1 F F F K K K T i T_i T i H ⊇ K ≤ e H \supseteq K_{\leq e} H ⊇ K ≤ e F F F e e e F F F K → H K \to H K → H K ≤ e K_{\leq e} K ≤ e q q q H H H D D D Q Q Q
For g = 1 g = 1 g = 1 F = k [ x ] F = k[x] F = k [ x ] D D D K K K k [ x ] k[x] k [ x ] K ≅ k [ x ] r K \cong k[x]^r K ≅ k [ x ] r 0 ≠ q = ( q 1 , … , q r ) 0 \ne q = (q_1,\ldots,q_r) 0 = q = ( q 1 , … , q r ) q j ≠ 0 q_j \ne0 q j = 0 p p p q j q_j q j k [ x ] k[x] k [ x ] p p p K → k [ x ] / ( p ) K \to k[x]/(p) K → k [ x ] / ( p ) q q q g = 0 g = 0 g = 0 F = k F = k F = k
For Q = F n / N Q = F^n/N Q = F n / N F n F^n F n N N N ⇒ \Rightarrow ⇒ ⇒ \Rightarrow ⇒ Q Q Q ⇔ \Leftrightarrow ⇔
Remark 5.9 (Finite and infinite intersections). Theorem 5.8 gives a finite intersection of prime submodules, but the intersection of finitely generated maximal submodules may have to be infinite. Indeed, a finite such intersection would embed F n / N F^n/N F n / N F n / N F^n/N F n / N g ≥ 1 g \geq1 g ≥ 1 F F F
Prime radicals
Corollary 5.10 (Finite generation of the prime radical). Let N N N F n F^n F n rad N \operatorname{rad} N rad N N N N rad N / N \operatorname{rad} N/N rad N / N k k k rad N \operatorname{rad} N rad N
rad N = ⋂ { P ⊆ F n : P is a prime submodule and N ⊆ P } . \operatorname{rad} N=\bigcap\{P\subseteq F^n:P\text{ is a prime submodule and }N\subseteq P\}. rad N = ⋂ { P ⊆ F n : P is a prime submodule and N ⊆ P } . Proof. Intersections of semiprime submodules are semiprime by Lemma 2.4 (2), so rad N \operatorname{rad} N rad N Q = F n / N Q=F^n/N Q = F n / N A = rad N / N A=\operatorname{rad} N/N A = rad N / N Q Q Q [25 Theorem 2] gives a free submodule U ⊆ Q U\subseteq Q U ⊆ Q dim k ( Q / U ) < ∞ \dim_k(Q/U)<\infty dim k ( Q / U ) < ∞ U ≠ 0 U\ne0 U = 0 F F F K ⊆ Q K\subseteq Q K ⊆ Q K ∩ U = 0 K\cap U=0 K ∩ U = 0 U U U Q / K Q/K Q / K K K K U U U U U U Lemma 2.6 makes Q / K Q/K Q / K Lemma 2.4 (3), the inverse image of K K K F n F^n F n N N N A ⊆ K A\subseteq K A ⊆ K A ∩ U = 0 A\cap U=0 A ∩ U = 0 U = 0 U=0 U = 0 A A A Q / U Q/U Q / U k k k A A A F F F N N N rad N \operatorname{rad} N rad N Q / A = F n / rad N Q/A=F^n/\operatorname{rad} N Q / A = F n / rad N Theorem 5.2 expresses rad N \operatorname{rad} N rad N N N N Lemma 2.4 (1).
Theorem D, together with the evaluation description in Proposition 5.5 , gives the following form of Hilbert’s Nullstellensatz.
Corollary 5.11 (Hilbert’s Nullstellensatz). For a finitely generated submodule N ⊆ F n N\subseteq F^n N ⊆ F n
rad N = { p ∈ F n : ε π , v ( p ) = 0 whenever ε π , v ( N ) = 0 } . \operatorname{rad} N=\{p\in F^n:\varepsilon_{\pi,v}(p)=0\text{ whenever }\varepsilon_{\pi,v}(N)=0\}. rad N = { p ∈ F n : ε π , v ( p ) = 0 whenever ε π , v ( N ) = 0 } . where π \pi π F F F k k k v v v ( V π ) n (V_\pi)^n ( V π ) n k k k
Proof. By Proposition 5.5 , the evaluation kernels in the statement are maximal, hence semiprime by Lemma 2.4 (1). Thus every such kernel containing N N N rad N \operatorname{rad} N rad N rad N \operatorname{rad} N rad N Theorem 5.8 and Proposition 5.5 separate every p ∉ rad N p\notin\operatorname{rad} N p ∈ / rad N
Irreducibility is essential here: kernels of arbitrary finite-dimensional representations need not be semiprime.
Free algebras with involution
This section proves Theorem E, except for the algorithmic assertion recalled in Appendix § 7 .§ 2 . Part (1) follows from Theorem D(1), since real modules are semiprime and their submodules are real. Positive truncations and adjoint extension prove parts (2) and (3).
Let k k k F = k ⟨ x 1 , … , x g ⟩ F=k\langle x_1,\ldots,x_g\rangle F = k ⟨ x 1 , … , x g ⟩ g < ∞ g<\infty g < ∞ k k k k = R k=\mathbb{R} k = R k = C k=\mathbb{C} k = C ∗ * ∗
Real prime separation
Theorem 6.1 (Free real modules). Let k k k F = k ⟨ x 1 , … , x g ⟩ F=k\langle x_1,\ldots,x_g\rangle F = k ⟨ x 1 , … , x g ⟩ g < ∞ g<\infty g < ∞ k k k F F F N ⊆ M N\subseteq M N ⊆ M M / N M/N M / N N N N M M M M M M
Proof. A real module is semiprime by Lemma 3.6 , so Theorem 5.2 writes Q = M / N Q=M/N Q = M / N Q Q Q Lemma 3.6 . The kernels of the coordinate projections are real prime submodules with zero intersection; take inverse images in M M M M M M F r → M F^r \to M F r → M r r r H i H_i H i Q Q Q F r → M → H i F^r \to M \to H_i F r → M → H i M M M
The real Jacobson property
For the remainder of this section, k = R k=\mathbb{R} k = R k = C k=\mathbb{C} k = C x i ∗ = x σ ( i ) x_i^*=x_{\sigma(i)} x i ∗ = x σ ( i ) σ \sigma σ σ = id \sigma=\operatorname{id} σ = id ∗ * ∗ ∗ * ∗ F F F V V V π ( f ∗ ) = π ( f ) ∗ \pi(f^*)=\pi(f)^* π ( f ∗ ) = π ( f ) ∗ k = R k=\mathbb{R} k = R k = C k=\mathbb{C} k = C X i ∗ = X σ ( i ) X_i^*=X_{\sigma(i)} X i ∗ = X σ ( i ) V ‾ ⊗ F V → k \overline{V}\otimes_F V\to k V ⊗ F V → k u ⊗ w ↦ ⟨ u , w ⟩ u\otimes w\mapsto\langle u,w\rangle u ⊗ w ↦ ⟨ u , w ⟩ ∑ i u i ⊗ u i \sum_i u_i\otimes u_i ∑ i u i ⊗ u i ∑ i ∥ u i ∥ 2 \sum_i\lVert u_i\rVert^2 ∑ i ∥ u i ∥ 2 Lemma 4.15 .
Lemma 6.2 (Positive truncation). Let N ⊆ F n N\subseteq F^n N ⊆ F n Q = F n / N Q=F^n/N Q = F n / N e ≥ 1 e\geq1 e ≥ 1 Q ≤ e Q_{\leq e} Q ≤ e
⟨ x i u , w ⟩ = ⟨ u , x σ ( i ) w ⟩ ( u , w ∈ Q ≤ e − 1 ) . \langle x_i u,w\rangle=\langle u,x_{\sigma(i)}w\rangle\qquad(u,w\in Q_{\leq e-1}). ⟨ x i u , w ⟩ = ⟨ u , x σ ( i ) w ⟩ ( u , w ∈ Q ≤ e − 1 ) . Proof. Let Z ⊆ Q ‾ ⊗ F Q Z\subseteq\overline{Q}\otimes_F Q Z ⊆ Q ⊗ F Q u ⊗ w u\otimes w u ⊗ w u , w ∈ Q ≤ e u,w\in Q_{\leq e} u , w ∈ Q ≤ e Q ≤ e Q_{\leq e} Q ≤ e Σ = { u ⊗ u : ∥ u ∥ = 1 } \Sigma=\{u\otimes u:\lVert u\rVert=1\} Σ = { u ⊗ u : ∥ u ∥ = 1 } 0 0 0 ∑ i λ i u i ⊗ u i = 0 \sum_i\lambda_i u_i\otimes u_i=0 ∑ i λ i u i ⊗ u i = 0 λ i ≥ 0 \lambda_i\geq0 λ i ≥ 0 ∑ λ i = 1 \sum\lambda_i=1 ∑ λ i = 1 ∑ i ( λ i u i ) ⊗ ( λ i u i ) = 0 \sum_i(\sqrt{\lambda_i}u_i)\otimes(\sqrt{\lambda_i}u_i)=0 ∑ i ( λ i u i ) ⊗ ( λ i u i ) = 0 λ i u i = 0 \sqrt{\lambda_i}u_i=0 λ i u i = 0 φ \varphi φ Z Z Z Σ \Sigma Σ τ ( u ⊗ w ) = w ⊗ u \tau(u\otimes w)=w\otimes u τ ( u ⊗ w ) = w ⊗ u Q ‾ ⊗ F Q \overline{Q}\otimes_F Q Q ⊗ F Q Z Z Z Σ \Sigma Σ φ \varphi φ ( φ + φ ∘ τ ) / 2 (\varphi+\varphi\circ\tau)/2 ( φ + φ ∘ τ ) /2
For k = R k=\mathbb{R} k = R ⟨ u , w ⟩ = φ ( u ⊗ w ) \langle u,w\rangle=\varphi(u\otimes w) ⟨ u , w ⟩ = φ ( u ⊗ w ) k = C k=\mathbb{C} k = C ⟨ u , w ⟩ = φ ( u ⊗ w ) − i φ ( u ⊗ i w ) \langle u,w\rangle=\varphi(u\otimes w)-i\varphi(u\otimes iw) ⟨ u , w ⟩ = φ ( u ⊗ w ) − i φ ( u ⊗ i w ) ( λ u ) ⊗ w = u ⊗ λ ‾ w (\lambda u)\otimes w=u\otimes\overline{\lambda}w ( λ u ) ⊗ w = u ⊗ λ w φ ∘ τ = φ \varphi\circ\tau=\varphi φ ∘ τ = φ ⟨ u , u ⟩ = φ ( u ⊗ u ) > 0 \langle u,u\rangle=\varphi(u\otimes u)>0 ⟨ u , u ⟩ = φ ( u ⊗ u ) > 0 u ≠ 0 u\neq0 u = 0 x i u ‾ = u ‾ ⋅ x i ∗ \overline{x_i u}=\overline{u}\cdot x_i^* x i u = u ⋅ x i ∗ Q ‾ \overline{Q} Q x i u ‾ ⊗ w = u ‾ ⊗ x σ ( i ) w \overline{x_i u}\otimes w=\overline{u}\otimes x_{\sigma(i)}w x i u ⊗ w = u ⊗ x σ ( i ) w u , w ∈ Q ≤ e − 1 u,w\in Q_{\leq e-1} u , w ∈ Q ≤ e − 1 Z Z Z
:::
Lemma 6.3 (Adjoint extension). Let V V V V ′ ⊆ V V'\subseteq V V ′ ⊆ V E E E T i : V ′ → V T_i:V'\to V T i : V ′ → V ⟨ T i u , w ⟩ = ⟨ u , T σ ( i ) w ⟩ \langle T_i u,w\rangle=\langle u,T_{\sigma(i)}w\rangle ⟨ T i u , w ⟩ = ⟨ u , T σ ( i ) w ⟩ u , w ∈ V ′ u,w\in V' u , w ∈ V ′
X i = T i E + ( T σ ( i ) E ) ∗ ( 1 − E ) X_i=T_iE+(T_{\sigma(i)}E)^*(1-E) X i = T i E + ( T σ ( i ) E ) ∗ ( 1 − E ) satisfies X i ∣ V ′ = T i X_i|_{V'}=T_i X i ∣ V ′ = T i X i ∗ = X σ ( i ) X_i^*=X_{\sigma(i)} X i ∗ = X σ ( i )
Proof. Set A i = T i E ∈ End ( V ) A_i=T_iE\in\operatorname{End}(V) A i = T i E ∈ End ( V ) E u , E w Eu,Ew E u , E w A i ∗ E = E A σ ( i ) A_i^*E=EA_{\sigma(i)} A i ∗ E = E A σ ( i ) V ′ V' V ′ X i = A i + A σ ( i ) ∗ ( 1 − E ) X_i=A_i+A_{\sigma(i)}^*(1-E) X i = A i + A σ ( i ) ∗ ( 1 − E ) X i X_i X i T i T_i T i
X σ ( i ) − X i ∗ = E A σ ( i ) − A i ∗ E = 0. X_{\sigma(i)}-X_i^*=EA_{\sigma(i)}-A_i^*E=0. X σ ( i ) − X i ∗ = E A σ ( i ) − A i ∗ E = 0. All adjoints here are adjoints of endomorphisms of V V V
Theorem 6.4 (The real Jacobson property). Every finitely generated real submodule N ⊆ F n N\subseteq F^n N ⊆ F n ε π , v \varepsilon_{\pi,v} ε π , v ∗ * ∗
Proof. Fix q ∈ F n ∖ N q\in F^n\setminus N q ∈ F n ∖ N e ≥ 1 e\geq1 e ≥ 1 q q q N N N Q = F n / N Q=F^n/N Q = F n / N V = Q ≤ e V=Q_{\leq e} V = Q ≤ e T i : Q ≤ e − 1 → V T_i:Q_{\leq e-1}\to V T i : Q ≤ e − 1 → V X i X_i X i V V V X i ∗ = X σ ( i ) X_i^\ast= X_{\sigma(i)} X i ∗ = X σ ( i ) F F F V V V ∗ \ast ∗
With v j = e j + N ∈ V v_j=e_j+N\in V v j = e j + N ∈ V F F F
ψ : F n ⟶ V , ψ ( f 1 , … , f n ) = ∑ j f j ( X ) v j . \psi:F^n\longrightarrow V,\qquad\psi(f_1,\ldots,f_n)=\sum_j f_j(X)v_j. ψ : F n ⟶ V , ψ ( f 1 , … , f n ) = j ∑ f j ( X ) v j . By Lemma 5.6 , ψ ( p ) = p + N \psi(p)=p+N ψ ( p ) = p + N deg p ≤ e \deg p\le e deg p ≤ e ψ \psi ψ N N N N N N ψ ( q ) = q + N ≠ 0 \psi(q)=q+N\ne0 ψ ( q ) = q + N = 0 V V V p p p
By Lemma 4.15 , V V V ∗ \ast ∗ F F F S S S ψ ( q ) \psi(q) ψ ( q ) F n ↠ V ↠ S F^n\twoheadrightarrow V\twoheadrightarrow S F n ↠ V ↠ S L L L N N N q q q S S S L L L Proposition 5.5 . As q q q N N N
The construction needs neither Lewin’s theorem nor the completion lemma: positivity makes the extension problem trivial.
Corollary 6.5 (Real maximal kernels). For a real maximal submodule L ⊂ F n L\subset F^n L ⊂ F n L = L π , v L=L_{\pi,v} L = L π , v ∗ \ast ∗ π \pi π v ≠ 0 v\ne0 v = 0 Proposition 5.5 .
Proof. By Proposition 5.5 , finite generation and finite codimension are equivalent. Under these conditions, apply Theorem 6.4 to L L L L L L L L L
Example 6.6 (Weak reality and the Jacobson property). Let F = R ⟨ x , x ∗ ⟩ F=\mathbb{R}\langle x,x^\ast\rangle F = R ⟨ x , x ∗ ⟩ p = x ∗ x + x − x ∗ p=x^\ast x+x-x^\ast p = x ∗ x + x − x ∗ N = F p N=Fp N = F p N N N
Proof. On J = F x ⊕ F x ∗ J=Fx\oplus Fx^\ast J = F x ⊕ F x ∗ F F F ϕ ( a x + b x ∗ ) = a + b ( x ∗ + 1 ) \phi(ax+bx^\ast)=a+b(x^\ast+1) ϕ ( a x + b x ∗ ) = a + b ( x ∗ + 1 ) ϕ ( p ) = 0 \phi(p)=0 ϕ ( p ) = 0 N ⊆ ker ϕ N\subseteq\ker\phi N ⊆ ker ϕ a + b ( x ∗ + 1 ) = 0 a+b(x^\ast+1)=0 a + b ( x ∗ + 1 ) = 0 a x + b x ∗ = − b p ax+bx^\ast=-bp a x + b x ∗ = − b p ker ϕ = N \ker\phi=N ker ϕ = N ∑ i f i ∗ f i ∈ N \sum_i f_i^\ast f_i\in N ∑ i f i ∗ f i ∈ N f i f_i f i f i = a i x + b i x ∗ f_i=a_i x+b_i x^\ast f i = a i x + b i x ∗ u i = ϕ ( f i ) u_i=\phi(f_i) u i = ϕ ( f i ) F F F ϕ \phi ϕ ∑ i f i ∗ f i ∈ N = ker ϕ \sum_i f_i^\ast f_i\in N=\ker\phi ∑ i f i ∗ f i ∈ N = ker ϕ
0 = ∑ i f i ∗ u i = x ∗ ∑ i a i ∗ u i + x ∑ i b i ∗ u i . 0=\sum_i f_i^\ast u_i=x^\ast\sum_i a_i^\ast u_i+x\sum_i b_i^\ast u_i. 0 = i ∑ f i ∗ u i = x ∗ i ∑ a i ∗ u i + x i ∑ b i ∗ u i . Comparing first letters shows that both coefficient sums vanish, so
∑ i u i ∗ u i = ∑ i a i ∗ u i + ( x + 1 ) ∑ i b i ∗ u i = 0. \sum_i u_i^\ast u_i=\sum_i a_i^\ast u_i+(x+1)\sum_i b_i^\ast u_i=0. i ∑ u i ∗ u i = i ∑ a i ∗ u i + ( x + 1 ) i ∑ b i ∗ u i = 0. The free algebra is formally real, as comparison of highest-degree coefficients shows. Hence every u i = 0 u_i=0 u i = 0 f i ∈ N f_i\in N f i ∈ N p + p ∗ = 2 x ∗ x p+p^\ast=2x^\ast x p + p ∗ = 2 x ∗ x x ∉ N x\notin N x ∈ / N N N N N N N Proposition 3.5 , so Theorem 5.8 gives the asserted intersection of maximal left ideals; an intersection of real left ideals would be real. □
Real radicals
Let N ⊂ F n N\subset F^n N ⊂ F n N r e \sqrt[\mathrm{re}]{N} re N § 3 ) need not have finite codimension over N N N Example 6.7 ), but it is finitely generated. Throughout, g g g ∗ \ast ∗
Example 6.7 (Comparison with the prime radical). rad N / N \operatorname{rad}N/N rad N / N N r e / N \sqrt[\mathrm{re}]{N}/N re N / N R ⟨ x , y ⟩ \mathbb{R}\langle x,y\rangle R ⟨ x , y ⟩ N = F x 2 N=Fx^2 N = F x 2 x ∈ N r e x\in\sqrt[\mathrm{re}]{N} x ∈ re N y j x y^j x y j x j ≥ 0 j\ge0 j ≥ 0 N N N
Theorem 6.8 (Finite generation of the real radical). Let N ⊆ F n N \subseteq F^n N ⊆ F n d ≥ 1 d \ge1 d ≥ 1
N r e = N + F ( N r e ∩ F < d n ) . \sqrt[\mathrm{re}]{N}=N+F\left(\sqrt[\mathrm{re}]{N}\cap F^n_{<d}\right). re N = N + F ( re N ∩ F < d n ) . where F < d n F^n_{<d} F < d n d d d N r e \sqrt[\mathrm{re}]{N} re N n ∑ j = 0 d − 1 g j n\sum_{j=0}^{d-1}g^j n ∑ j = 0 d − 1 g j d d d
Proof. Put M = F 1 × n M=F^{1\times n} M = F 1 × n R = N r e R=\sqrt[\mathrm{re}]{N} R = re N J = N + F ( R ∩ M < d ) J=N+F(R\cap M_{<d}) J = N + F ( R ∩ M < d ) N ⊆ J ⊆ R N\subseteq J\subseteq R N ⊆ J ⊆ R J J J ≤ d \le d ≤ d J ≠ R J\ne R J = R p ∈ R ∖ J p\in R\setminus J p ∈ R ∖ J e e e e ≥ d e\ge d e ≥ d
V = ( M / J ) ≤ e , U = ( M / J ) ≤ e − 1 , W = ( M / R ) ≤ e . V=(M/J)_{\le e},\qquad U=(M/J)_{\le e-1},\qquad W=(M/R)_{\le e}. V = ( M / J ) ≤ e , U = ( M / J ) ≤ e − 1 , W = ( M / R ) ≤ e . and let α : V → W \alpha:V\to W α : V → W e e e J J J R R R < e <e < e U ∩ ker α = 0 U\cap\ker\alpha=0 U ∩ ker α = 0 ∗ * ∗ p + J p+J p + J p ∈ R p\in R p ∈ R
By Lemma 6.2, applied to the real submodule R R R W W W ⟨ x i u , w ⟩ = ⟨ u , x σ ( i ) w ⟩ \langle x_i u,w\rangle=\langle u,x_{\sigma(i)}w\rangle ⟨ x i u , w ⟩ = ⟨ u , x σ ( i ) w ⟩ u , w ∈ ( M / R ) ≤ e − 1 u,w\in(M/R)_{\le e-1} u , w ∈ ( M / R ) ≤ e − 1 V V V ker α \ker\alpha ker α S ⊇ U S\supseteq U S ⊇ U ker α \ker\alpha ker α V = S ⊕ ker α V=S\oplus\ker\alpha V = S ⊕ ker α α ∣ S \alpha|_S α ∣ S ker α \ker\alpha ker α ker α \ker\alpha ker α U U U U U U
⟨ v , u ⟩ V = ⟨ α v , α u ⟩ W ( v ∈ V , u ∈ U ) . \langle v,u\rangle_V=\langle\alpha v,\alpha u\rangle_W\qquad(v\in V,\ u\in U). ⟨ v , u ⟩ V = ⟨ α v , α u ⟩ W ( v ∈ V , u ∈ U ) . Since α \alpha α T i : U → V T_i:U\to V T i : U → V u ↦ x i u u\mapsto x_i u u ↦ x i u
⟨ T i u , w ⟩ V = ⟨ x i α u , α w ⟩ W = ⟨ α u , x σ ( i ) α w ⟩ W = ⟨ u , T σ ( i ) w ⟩ V ( u , w ∈ U ) . \langle T_i u,w\rangle_V=\langle x_i\alpha u,\alpha w\rangle_W=\langle\alpha u,x_{\sigma(i)}\alpha w\rangle_W=\langle u,T_{\sigma(i)}w\rangle_V\qquad(u,w\in U). ⟨ T i u , w ⟩ V = ⟨ x i α u , α w ⟩ W = ⟨ α u , x σ ( i ) α w ⟩ W = ⟨ u , T σ ( i ) w ⟩ V ( u , w ∈ U ) . Lemma 6.3 extends the T i T_i T i X i X_i X i V V V X i ∗ = X σ ( i ) X_i^*=X_{\sigma(i)} X i ∗ = X σ ( i ) V V V F F F x i ⋅ v = X i v x_i\cdot v=X_i v x i ⋅ v = X i v M / J M/J M / J U U U V V V M / J M/J M / J V ⊗ ‾ F V → k V\overline{\otimes}_F V\to k V ⊗ F V → k v ‾ ⊗ w ↦ ⟨ v , w ⟩ V \overline{v}\otimes w\mapsto\langle v,w\rangle_V v ⊗ w ↦ ⟨ v , w ⟩ V ∑ i v i ‾ ⊗ v i = 0 \sum_i\overline{v_i}\otimes v_i=0 ∑ i v i ⊗ v i = 0 ∑ i ∥ v i ∥ 2 = 0 \sum_i\lVert v_i\rVert^2=0 ∑ i ∥ v i ∥ 2 = 0 v i = 0 v_i=0 v i = 0
Since all generators of J J J ≤ d ≤ e \le d\le e ≤ d ≤ e F F F ψ : M / J → V \psi:M/J\to V ψ : M / J → V V V V f : M → M / J → ψ V f:M\to M/J\xrightarrow{\psi}V f : M → M / J ψ V V V V ker f \ker f ker f N N N R R R p ∈ R p\in R p ∈ R f ( p ) = 0 f(p)=0 f ( p ) = 0 f ( p ) = ψ ( p + J ) = p + J ≠ 0 f(p)=\psi(p+J)=p+J\ne0 f ( p ) = ψ ( p + J ) = p + J = 0 ψ \psi ψ V V V J = R J=R J = R R ∩ M < d R\cap M_{<d} R ∩ M < d dim k M < d = n ∑ j = 0 d − 1 g j \dim_k M_{<d}=n\sum_{j=0}^{d-1}g^j dim k M < d = n ∑ j = 0 d − 1 g j
If N N N d = 0 d=0 d = 0 k n k^n k n F n / N F^n/N F n / N N r e = N \sqrt[\mathrm{re}]{N}=N re N = N
Remark 6.9 (Relation to earlier finite-generation results). The finite-generation and degree-bound conclusions of Theorem 6.8 are known: see [8 Corollary 2.6 and Theorem 3.1] for left ideals, and [19 Proposition 2.4] and [30 Theorem 7.3] for row modules. The latter results imply that the real radical is generated by N N N < d <d < d § 7 .§ 2 .
Theorem E, together with the evaluation description in Corollary 6.5 , gives the following real Nullstellensatz.
Corollary 6.10 (Real Nullstellensatz). For a finitely generated submodule N ⊆ F n N\subseteq F^n N ⊆ F n
N r e = { p ∈ F n : ε π , ν ( p ) = 0 whenever ε π , ν ( N ) = 0 } , \sqrt[\mathrm{re}]{N}=\{p\in F^n:\varepsilon_{\pi,\nu}(p)=0\text{ whenever }\varepsilon_{\pi,\nu}(N)=0\}, re N = { p ∈ F n : ε π , ν ( p ) = 0 whenever ε π , ν ( N ) = 0 } , where π \pi π ∗ * ∗ ν \nu ν
Proof. Each ker ε π , v \ker\varepsilon_{\pi,v} ker ε π , v N N N N r e \sqrt[\mathrm{re}]{N} re N N r e \sqrt[\mathrm{re}]{N} re N p ∉ N r e p \notin\sqrt[\mathrm{re}]{N} p ∈ / re N
Remark 6.11 (Relation to earlier real Nullstellensätze). For k = R k = \mathbb{R} k = R [19 Corollary 1.11] prove Corollary 6.10 for k ⟨ x 1 , x 1 ∗ , … , x g , x g ∗ ⟩ k\langle x_1,x_1^*,\ldots,x_g,x_g^*\rangle k ⟨ x 1 , x 1 ∗ , … , x g , x g ∗ ⟩ k ⟨ x 1 , … , x g ⟩ k\langle x_1,\ldots,x_g\rangle k ⟨ x 1 , … , x g ⟩ x i ∗ = x i x_i^*=x_i x i ∗ = x i [8 Theorem 1.6] and for row modules in [30 Theorem 1.3] .
We now show how an involution permuting the generators reduces to the first case. An involutive permutation consists of disjoint transpositions and fixed points. After renaming, g = 2 r + s g=2r+s g = 2 r + s
F = k ⟨ a 1 , a 1 ∗ , … , a r , a r ∗ , y 1 , … , y s ⟩ , y j ∗ = y j . F=k\langle a_1,a_1^*,\ldots,a_r,a_r^*,y_1,\ldots,y_s\rangle,\qquad y_j^*=y_j. F = k ⟨ a 1 , a 1 ∗ , … , a r , a r ∗ , y 1 , … , y s ⟩ , y j ∗ = y j . Replacing each y j y_j y j z j , z j ∗ z_j,z_j^* z j , z j ∗ ∗ * ∗ B B B ∗ * ∗ ι : F → B \iota:F\to B ι : F → B ρ : B → F \rho:B\to F ρ : B → F
ι ( y j ) = ( z j + z j ∗ ) / 2 , ρ ( z j ) = ρ ( z j ∗ ) = y j , \iota(y_j)=(z_j+z_j^*)/2,\qquad\rho(z_j)=\rho(z_j^*)=y_j, ι ( y j ) = ( z j + z j ∗ ) /2 , ρ ( z j ) = ρ ( z j ∗ ) = y j , and fixing every a i , a i ∗ a_i,a_i^* a i , a i ∗ ρ ι = 1 F \rho\iota=1_F ρ ι = 1 F N ⊆ F n N\subseteq F^n N ⊆ F n J = B ι ( N ) J=B\iota(N) J = B ι ( N ) ρ − 1 ( N r e ) \rho^{-1}(\sqrt[\mathrm{re}]{N}) ρ − 1 ( re N ) J J J p ∉ N r e p\notin\sqrt[\mathrm{re}]{N} p ∈ / re N ι ( p ) ∉ J r e \iota(p)\notin\sqrt[\mathrm{re}]{J} ι ( p ) ∈ / re J [19 Corollary 1.11] , choose a finite-dimensional ∗ * ∗ ψ \psi ψ v v v J J J ι ( p ) \iota(p) ι ( p ) π = ψ ∘ ι \pi=\psi\circ\iota π = ψ ∘ ι ε π , v = ε ψ , v ∘ ι \varepsilon_{\pi,v}=\varepsilon_{\psi,v}\circ\iota ε π , v = ε ψ , v ∘ ι N N N p p p
Our proofs construct separating representations directly on quotient truncations using Lemmas 6.2 and 6.3. The same lemmas give the radical degree bound, without extending a positive functional to all of F F F
Computing radicals over free algebras
This appendix proves the algorithmic assertion in Theorem D(3), making Corollary 5.10 effective. The procedure uses one-sided normal forms [29 Section 2.3] , [17 ] , and radicals of finite-dimensional algebras [14 Sections 2.3.1–2.3.2] . Throughout, F = k ⟨ x 1 , … , x g ⟩ F=k\langle x_1,\ldots,x_g\rangle F = k ⟨ x 1 , … , x g ⟩ g ≥ 1 g\geq1 g ≥ 1 N ⊆ F 1 × n N\subseteq F^{1\times n} N ⊆ F 1 × n ≤ d \leq d ≤ d d ≥ 1 d\geq1 d ≥ 1 D = n ∑ j = 0 d − 1 g j D=n\sum_{j=0}^{d-1}g^j D = n ∑ j = 0 d − 1 g j
Normal forms and the prime-radical calculation
Fix a degree-lexicographic order on row monomials. Left reduction of the input generators gives a finite left Gröbner basis of degrees ≤ d \leq d ≤ d ≥ d \geq d ≥ d d d d V V V < d <d < d
Proposition A.1 (The finite-dimensional calculation). Let B B B Q = F n / N Q=F^n/N Q = F n / N B = 0 B=0 B = 0 rad N = N \operatorname{rad}N=N rad N = N E E E S = ker ( F → E / J ( E ) ) S=\ker(F\to E/J(E)) S = ker ( F → E / J ( E ))
rad N / N = B ∩ S Q . \operatorname{rad}N/N=B\cap SQ. rad N / N = B ∩ S Q . Proof. The sum of all finite-dimensional submodules lies in V V V B B B A = rad N / N A=\operatorname{rad}N/N A = rad N / N A ⊆ B A\subseteq B A ⊆ B Q / S Q Q/SQ Q / S Q F / S F/S F / S S Q SQ S Q F n F^n F n rad N \operatorname{rad}N rad N A ⊆ S Q A\subseteq SQ A ⊆ S Q B / A B/A B / A Q / A Q/A Q / A
J ( E ) J(E) J ( E ) J ( E ) J(E) J ( E ) B / A B/A B / A S S S Q / A Q/A Q / A ( B / A ) ∩ S ( Q / A ) = 0 (B/A) \cap S(Q/A) = 0 ( B / A ) ∩ S ( Q / A ) = 0 B ∩ S Q = A B \cap SQ = A B ∩ S Q = A
Theorem A.2 (Computing the prime radical). Over a field with effective arithmetic and an algorithm for radicals of finite-dimensional algebras, the following procedure computes rad N \operatorname{rad} N rad N D D D < d < d < d Q \mathbb{Q} Q
Proof. All module operations use the normal forms just described.
Starting with V 0 = V V_0 = V V 0 = V V r + 1 = { v ∈ V r : x i v ∈ V r for all i } V_{r+1} = \{v \in V_r : x_i v \in V_r \text{ for all } i\} V r + 1 = { v ∈ V r : x i v ∈ V r for all i } D D D B B B V V V B = 0 B = 0 B = 0
Compute the algebra E E E B B B J ( E ) J(E) J ( E ) [14 Sections 2.3.1–2.3.2] . The algebra-generation calculation takes place in End k ( B ) \operatorname{End}_k(B) End k ( B ) D 2 D^2 D 2
Compute left generators of S = ker ( F → E / J ( E ) ) S = \ker(F \to E/J(E)) S = ker ( F → E / J ( E )) w 1 = 1 , … , w t w_1 = 1, \ldots, w_t w 1 = 1 , … , w t E / J ( E ) E/J(E) E / J ( E ) x i w j = ∑ ℓ c i j ℓ w ℓ x_i w_j = \sum_\ell c_{ij\ell}w_\ell x i w j = ∑ ℓ c ij ℓ w ℓ s i j = x i w j − ∑ ℓ c i j ℓ w ℓ s_{ij} = x_i w_j - \sum_\ell c_{ij\ell}w_\ell s ij = x i w j − ∑ ℓ c ij ℓ w ℓ S S S
Reduce a basis b 1 , … , b s b_1, \ldots, b_s b 1 , … , b s B B B < d < d < d K = N + S F n K = N + SF^n K = N + S F n k s → F n / K k^s \to F^n/K k s → F n / K ( a j ) ↦ ∑ j a j b j + K (a_j) \mapsto\sum_j a_j b_j + K ( a j ) ↦ ∑ j a j b j + K B ∩ S Q B \cap SQ B ∩ S Q
Proposition A.1 proves correctness. There are at most dim k B ≤ D \dim_k B \le D dim k B ≤ D < d < d < d
The real-radical algorithm
For k = R k = \mathbb{R} k = R k = C k = \mathbb{C} k = C [8 Section 3.1] , in the row-module formulation of [19 Section 7.5 and Proposition 7.11] , computes N r e \sqrt[\mathrm{re}]{N} re N < d < d < d L ∩ F < d n L \cap F_{<d}^n L ∩ F < d n D D D Theorem 6.8 . With exact semidefinite feasibility, the procedure is effective for real algebraic coefficients, or complex coefficients with real algebraic real and imaginary parts; the feasibility conditions are polynomial equalities and inequalities over the real algebraic numbers.
If d = 0 d = 0 d = 0 N N N N N N g = 0 g = 0 g = 0 k k k
Acknowledgements
In the preparation of this paper the author used the AI assistants Claude (Anthropic) and ChatGPT (OpenAI) for conceptual discussions regarding proofs and literature, and for editing assistance. All arguments and references were independently verified by the author, who takes full responsibility for the content.
University of Ljubljana, Faculty of Mathematics and Physics, Department of Mathematics, Jadranska 21, SI-1000 Ljubljana, Slovenia, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia
Email address: jaka.cimpric@fmf.uni-lj.si
