LINK CLOSED SETS OF PRIME IDEALS AND STABILITY ON BIMODULES

Over a Noetherian ring R, each of a weakly symmetric pair of torsion radicals is shown to be stable on Noetherian R-R bimodules if and only if the set of prime ideals that are closed with respect to each torsion radical is closed under links. Such a pair for R, termed a weakly symmetric stable pair, is extended to a weakly symmetric stable pair for any Noetherian extension ring of R. In case classical Krull dimension is a link invariant, we give a positive answer to the incomparability question for linked prime ideals in certain extension rings of R. Mathematics Subject Classification (2010): 16D20, 16D25, 16S99


Introduction
For a ring R, a torsion radical is a hereditary (left exact) functor defined on the category R-Mod or Mod-R of left or right R-modules as given in [4] or [10].A pair of torsion radicals is an ordered pair (σ, τ ) where σ is a torsion radical defined on R-Mod and τ is a torsion radical defined on Mod-R.A pair (σ, τ ) is called weakly symmetric if σ(R/P ) = τ (R/P ) for all prime ideals P .In this case, Sp(σ, τ ) denotes the set of all σ-dense (τ -dense) prime ideals of R. A pair (σ, τ ) is called stable provided, for any ring Λ, σ and τ are stable on Noetherian R-Λ bimodules and Noetherian Λ-R bimodules respectively.The aim of first part of this paper is to show that when all rings involved are two-sided Noetherian, a weakly symmetric stable pair can be characterized internally as one for which the set of dense prime ideals Sp(σ, τ ) is closed under links.
The rest of the paper examines the problem of lifting a stable pair to an extension ring S ⊇ R with the idea of generating a link-closed set of prime ideals in S starting from a link-closed set of prime ideals in R. In particular, it is shown that if S ⊇ R are both Noetherian and (σ, τ ) is a weakly symmetric stable pair for R, then the canonical extension (σ * , τ * ) is a weakly symmetric stable pair for S. Thus, a link closed set of prime ideals of R of the form Sp(σ, τ ) gives rise to a link closed set Sp(σ * , τ * ) of prime ideals of S. We use this result to show that if R is a Noetherian ring where classical Krull dimension is a link invariant, then for linked prime ideals P, Q of S, R/(P ∩ R) and R/(Q ∩ R) have equal classical Krull dimension.As a corollary, linked prime ideals in a Noetherian extension ring S are incomparable provided S is either a finite normalizing extension of R or a strongly G-graded ring with base ring R for some finite group G.

Notation and definitions
All rings considered have an identity and modules are unitary.Properties and adjectives, when not accompanied by one-handed modifiers are meant to hold on both sides.However, all modules will be left R-modules unless otherwise specified.

The notation
in M , then we will write N ≤ e M .In case M is an S-R bimodule, we will write then ann M (X) = {m ∈ M | xm = 0 for all x ∈ X}.A module M is said to be finitely annihilated if there exist finitely many elements m 1 , . . ., m k ∈ M such that l R (M ) = l R (m 1 , . . ., m k ).Note that if M is a finitely annihilated module, then there is a one-to-one map R/l R (M ) → M (n) for some n.
The collection of all prime ideals of R is denoted by Spec(R).If U is a uniform R-module, and R is left Noetherian, then it follows from [5,Lemma 4.22] that there is a unique prime ideal P such that all nonzero X ≤ ann U (P ) have l R (X) = P .The ideal P is called the associated prime ideal (or assassinator) of U and is denoted by ass(U ).For an arbitrary module M , the set of associated prime ideals of M , denoted by Ass(M ), is the set of all prime ideals ass(U ) where U is a uniform submodule of M .The set of maximal members of Ass(M ) will be denoted by maxAss(M ).
If P ∈ Spec(R) and σ is a torsion radical defined on R-Mod or Mod-R, then P is called σ-dense provided R/P is σ-torsion while P is called σ-closed provided R/P is σ-torsion-free.The set of all σ-dense prime ideals is denoted by Spec σ (R), and the set of all σ-closed prime ideals is denoted by Spec σ (R).It is well known that a prime ideal P of a left Noetherian ring R is either σ-dense or σ-closed in case σ is defined on R-Mod.

Stability and linked primes
Throughout this section, R will denote a two-sided Noetherian ring.If C is a class of R-Λ bimodules for some ring Λ and σ is a torsion radical defined on R-mod, then we say that σ is stable on C provided that for all M ∈ C, σ(M ) is an essentially closed left R-submodule of M .Equivalently, σ is stable on C provided that for all Again, the right-hand version is defined in the obvious way.
The aim of this section is to show that for a pair (σ, τ ) with Spec σ (R) = Spec τ (R), stability on Noetherian bimodules is equivalent to the closure of this class of prime ideals under links.For P, Q ∈ Spec(R), there is a link from Q to P , denoted Q ∼→ P provided there exists an ideal A of R with QP ⊆ A ⊂ Q ∩ P such that (Q ∩ P )/A is torsion-free as a left R/Q-module and as a right R/P -module (see [5]).
Proposition 3.1.Let (σ, τ ) be a pair of torsion radicals. ( (2) If τ is stable on ideal factors of R and if P ∈ Spec τ (R) and Q ∼→ P , then Proof.Let A be an ideal with QP ⊆ A ⊂ Q ∩ P such that (Q ∩ P )/A is torsionfree on both sides.Let I/A = 0 be a left ideal of R/A.If P I A, then 0 = where the last equality follows since (Q ∩ P )/A is torsion-free over R/P .In either case, (P/A) ∩ (I/A) = 0. Thus, P/A ≤ eR R/A.Note that Q/A ≤ e R/A R by the symmetric argument.
For (2), QP ⊆ A and P τ -dense implies that Q/A is τ -torsion.The conclusion follows as in (1).
A nonempty subset Ω of Spec(R) is right link closed provided P ∈ Ω whenever Below, we show that the converse is true and, in this case, stability extends to a much larger class of bimodules.
Lemma 3.2.Let R B Λ = 0 be a right Noetherian bimodule where Λ is any ring.
(1) Every left R-submodule of B is finitely annihilated.
(2) If σ is a torsion radical and σ(B) = B, then every annihilator ideal of B is σ-dense.
(3) If σ is a torsion radical and σ(B) = 0, then every annihilator ideal of B is σ-closed.
(4) Let P ∈ Ass( R B).Then ann B (P ) is a torsion-free left R/P -module iff P ∈ maxAss( R B). ( or ( , there is a one-to-one map R/A → F (k) for some k.As B is σ-torsion or σ-torsionfree, then respectively, so is . The conclusion follows.A pair of torsion radicals (σ, τ ) with Spec σ (R) = Spec τ (R) will be called weakly symmetric.In this case, we will denote the set of all σ-dense (τ -dense) prime ideals by Sp(σ, τ ).A pair of torsion radicals (σ, τ ) such that σ is stable on all Noetherian R-Λ bimodules and τ is stable on all Noetherian Λ-R bimodules for any ring Λ will be called a stable pair.
(2) σ and τ are stable on ideal factors of R.

Ring extensions
Let S ⊇ R be a ring extension and let σ be a torsion radical defined on R-mod.
From [2, Ch.II(4)], the set is an idempotent filter.This defines a torsion radical σ * on S-mod given by It is easy to see that for any left S-module M , σ * (M ) is the unique largest Ssubmodule of M that is σ torsion as a left R-module.Furthermore, if J ∈ D, then R ∩ J is σ-dense.However, a left ideal of S that contracts to a σ-dense left ideal of R is not necessarily a member of D. Note though that the situation is different for two sided ideals; see Lemma 4.2.
Below, we show that stability of σ on bimodules passes to σ * .
Theorem 4.1.Let S ⊇ R be a Noetherian extension of a Noetherian ring R and let σ be a torsion radical.If σ is stable on Noetherian R-Λ bimodules, then σ * is stable on Noetherian S-Λ bimodules.
Proof.Let B be a Noetherian S-Λ bimodule with σ * (B) ≤ eS B. We claim that Clearly, it suffices to show that every member of Ass( R B) is σ-dense.
Then there is a uniform submodule 0 = U ≤ R B such that all 0 = X ≤ ann U (P ) satisfy P = l R (X).Let V = ann U (P ).By Lemma 3.2(1), V is finitely annihilated as a left R-module, hence there is a one-to-one map R/P → V (m) for some m.By [5, Proposition 6.11], V must be torsion-free as an R/P -module.
We claim that D is a prime ideal of S. To this end, let I, J be ideals of S such Next, we show that a weakly symmetric stable pair (σ, τ ) for R extends to a stable pair for S. Let P ∈ Spec σ * (S).Since σ * (S/P ) = S/P , σ(S/P ) = S/P and so σ(R/(P ∩ R)) = R/(P ∩ R).Thus, if Q 1 , . . ., Q n denote all of the prime ideals of R that are minimal over P ∩ R, then each As rings, R/Q ∼ = T while R/P ∼ = S. Also, QP = 0 and Since XS is torsion-free as a left T -module and as a right S-module, Q ∼→ P .
is both right and left link-closed.Thus, from Proposition 3.1, if (σ, τ ) is a pair of torsion radicals with Spec σ (R) = Spec τ (R) and both σ and τ are stable on ideal factors of a Noetherian ring R, then Spec σ (R) is link closed.

Proposition 3 . 3 .
Let (σ, τ ) be a weakly symmetric pair.If Sp(σ, τ ) is link closed, then σ is stable on all Noetherian R-Λ bimodules and τ is stable on all Noetherian Λ-R bimodules.Proof.If suffices to prove the first statement since the second follows by the symmetric argument.Let R B Λ be a Noetherian bimodule with σ(B) ≤ eR B. Then Ass( R B) = Ass(σ(B)) and so by Lemma 3.2(2), every member of Ass( R B) is σdense.If σ(B) = B, then by Noetherian induction, we can assume that the result holds true for all proper bimodule factors of B. Pick P ∈ maxAss( R B).Then P is σ-dense and so ann B (P ) ⊆ σ(B) ⊂ B. If σ(B) = ann B (P ), then B/ ann B (P ) is σ-torsionfree.Pick Q ∈ Ass(B/ ann B (P )).Then Q is σ-closed by Lemma 3.2(3) .Also, by Lemma 3.2(5), Q ∈ Ass( R B) or Q ∼→ P .Clearly, the former fails and, by hypothesis, so does the latter.It follows that σ(B) ⊃ ann B (P ).If σ(B)/ ann B (P ) is an essential submodule of B/ ann B (P ), then the conclusion follows from the inductive hypothesis.Suppose then that σ(B) ∩ X = ann B (P ) for some submodule X ⊃ ann B (P ).Now, σ(X) = σ(B) ∩ X = ann B (P ) and so X/ ann B (P ) is σ-torsionfree.Choose Q ∈ Ass(X/ ann B (P )) ⊂ Ass(B/ ann B (P )).By Lemma 3.2(3), Q is σ-closed.However, by Lemma 3.2(5) again, Q ∈ Ass( R B) or Q ∼→ P .As before, both alternatives fail.Therefore, σ(B) = B.
that IJ ⊆ D and suppose that J D. Then JX = JSX = S(JX) = 0 and D ⊆ l S (JX).By the maximality of D, D = l S (JX).Since IJ ⊆ D = l S (SX), IJSX = IJX = 0 and so I ⊆ D. Therefore, D is a prime ideal of S. By Lemma 3.2(1), SX is finitely annihilated as a left S-module.Thus, there is a one-to-one S-module map S/D → (SX) (n) for some n.Restricting this map to a nonzero uniform left ideal of S/D yields an isomorphism onto a nonzero uniform F ≤ S SX which is clearly torsion-free over S/D.Since σ * (B) ≤ eS B, σ * (F ) = σ * (B) ∩ F = 0. Then F torsion-free over S/D forces D = l S (σ * (F )).By Lemma 3.2(2), D is σ * -dense.Then D = l S (SX) forces SX to be σ * -torsion.It follows that σ(X) = X.Now, from above, P = l R (X).Using Lemma 3.2(2) again, P is σ-dense.Therefore, σ(B) ≤ e R B, and so σ(B) = B. Since σ * (B) is the largest S-submodule of B that is σ-torsion, σ * (B) = B.

Lemma 4 . 2 .Corollary 4 . 3 .
Let σ be a torsion radical defined on R-mod and let S ⊇ R be a ring extension.Let I ⊆ S be an ideal.Then I is σ * -dense if and only if I ∩R is σ-dense.Proof.From the definition of σ * , the forward implication is obvious.For the reverse implication, let I ⊆ S be an ideal such that I ∩ R is σ-dense.Fix s ∈ S and define an epimorphism R/(I ∩ R) → (Rs + I)/I via right multiplication by s. (Note that this map is well defined since I is an ideal of S.) It follows that for all s ∈ S, (Rs + I)/I is σ-torsion whence S/I is σ-torsion as a left R-module.Therefore, σ * (S/I) = S/I.Let S ⊇ R be a Noetherian extension of a Noetherian ring R. Let (σ, τ ) be a weakly symmetric stable pair for R. Then (σ * , τ * ) is a weakly symmetric stable pair for S. Proof.By Theorem 4.1, σ * is stable on Noetherian S-Λ bimodules for any ring Λ.In particular, σ * is stable on ideal factors of S. By Proposition 3.1, Spec σ * (S) is right link closed.Similarly, Spec τ * (S) is left link closed.Thus, by Corollary 3.4, it suffices to show that Spec σ * (S) = Spec τ * (S).

2 ,τCorollary 4 . 4 .From [ 5 ,
* (S/P ) = S/P .Therefore, Spec σ * (S) ⊆ Spec τ * (S).The reverse inclusion follows by the symmetric argument.Using Corollary 3.4, we can view the last result as way of obtaining a link closed set of prime ideals in S from a link closed set of prime ideals in R. Let S ⊇ R be a Noetherian extension of a Noetherian ring R. Let (σ, τ ) be a weakly symmetric pair for R. If Sp(σ, τ ) is link closed, then Sp(σ * , τ * ) is link closed.5. Invariance of classical Krull dimension Throughout, Cldim(R/I) will denote the classical Krull dimension of R/I where I ⊆ R is an ideal.For a Noetherian ring R, classical Krull dimension is called a link invariant provided Cldim(R/Q) = Cldim(R/P ) for P, Q ∈ Spec(R) with Q ∼→ P .The goal of this section is to show that, in this case, if S is any Noetherian extension ring of R, then Cldim(R/(Q ∩ R)) = Cldim(R/(P ∩ R)) for all prime ideals P, Q ∈ Spec(S) with Q ∼→ P .Lemma 12.2], if β ≤ Cldim(R), then there exists a prime ideal P such that Cldim(R/P ) = β.Thus, for each ordinal β ≤ Cldim(R), the set Spec β (R) = {P ∈ Spec(R)| Cldim(R/P ) ≥ β} It follows that XS is idempotent.In fact, XS is the only nonzero prime ideal of T whence Cldim(T ) = 1.Note that Cldim(S) = 0. From [9, 3.15], T does not satisfy the second layer condition.Furthermore, by [6, Example 5.2.18],XS is torsion-free as a left T -module and as a right S-module.Consider the formal upper triangular matrix ring