Gorenstein rings and irreducible parameter ideals

Abstract

Abstract. Given a Noetherian local ring (R, m) it is shown that there exists an integer such that R is Gorenstein if and only if some system of parameters contained in m generates an irreducible ideal. We obtain as a corollary that R is Gorenstein if and only if every power of the maximal ideal contains an irreducible parameter ideal.

On the index of reducibility in Noetherian modules

“…It is well-known that a Noetherian local ring R with dim R = d is Gorenstein if and only if R is Cohen-Macaulay with the Cohen-Macaulay type r(R) = dim k Ext d (k, M)) = 1. Therefore the following result, which is the main result of [16,Theorem], is an immediate consequence of Theorem 5.2. Remark 5.4.…”

A study of quasi-Gorenstein rings

“…In Section 5, we prove the following quasi-Gorenstein version of [30] (see Section 5 for the definition of the limit closure). It is worth pointing out that [30, Proposition 2.3.…”

A Characterization of Gorenstein Rings and Grothendieck's Conjecture

“…, x d be a system of parameters in m which generates an irreducible ideal. By [11,Proposition 2.8], the map ϕ m,x is surjective. The result now follows from Lemma 2.1.…”

“…By employing a method of proof which is similar to that used in [11,Theorem 2.9], it is easy to see that x 1 , . .…”

“…Therefore, R is Gorenstein if and only if every parameter ideal is irreducible. Recently, in[11], Marley, Rogers and Sakurai proved that there exists an integer such that R is Gorenstein if and only if some parameter ideal contained in m is irreducible. They also showed that the integer identified in their result may be taken to be the least integer δ = δ(R) such that the canonical map…”