LINK CLOSED SETS OF PRIME IDEALS AND STABILITY ON BIMODULES

Karl A. Kosler
2015 International Electronic Journal of Algebra  
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
more » ... parability 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 LINK CLOSED SETS OF PRIME IDEALS AND STABILITY ON BIMODULES 189 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 N ≤ R M , or N ≤ M when R is understood, will be used to indicate that N is a (left) R-submodule of M . Similarly for N ≤ M R . If N ≤ M is essential in M , then we will write N ≤ e M . In case M is an S-R bimodule, we will write S M R and use N ≤ S M R to indicate that N is an S-R subbimodule of M . For a nonempty subset X of an R-module M , l R (X) = {r ∈ R | rx = 0 for all x ∈ X}. In case X is a nonempty subset of a left R-module, then r R (X) = {r ∈ R | xr = 0 for all x ∈ X}. If X is a nonempty subset of R and M is a left R-module, 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. P ⊂ l R (Z) ⊆ Q contradicting our choice of P . (5) The last result is [7, Proposition 3.1]. 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 ). 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.
doi:10.24330/ieja.266220 fatcat:ahj53uzgjzfl3g4g4l2x57vv5q