Rings with Canonical Reductions
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Rings with Canonical Reductions Mehran Rahimi1 Received: 30 June 2019 / Accepted: 24 January 2020 © Iranian Mathematical Society 2020
Abstract We study the class of one-dimensional Cohen–Macaulay local rings with canonical reductions, i.e., admit canonical ideals which are reductions of the maximal ideals, show that it contains the class of almost Gorenstein rings, and study characterizations for rings obtained by idealizations or by numerical semigroup rings to have canonical reductions. Keywords Cohen–Macaulay local ring · Gorenstein ring · Almost Gorenstein ring · Canonical ideal · Canonical reduction Mathematics Subject Classification 13H10 · 13H15
1 Introduction In [1], Barucci and Fröberg, (1997) introduced the class of almost Gorenstein rings which is a subclass of one-dimensional analytically unramified rings. This concept has been generalized to Cohen–Macaulay local rings, studied in details by Goto et al. [11,13,14]. The theory is still developing (see [3,10]) introducing new classes of rings such as 2-almost Gorenstein local (2-AGL for short) and generalized Gorenstein local (GGL for short) rings. Throughout, (R, m, k) is a Noetherian local ring with maximal ideal m and all R-modules are finitely generated. Set e 0m (R) as the multiplicity of R with respect to m. For an R-module M, we denote (M) for the length of M as an R-module, and dim(M) (R/m, M)). If I and set the Cohen–Macaulay type of M as r (M) := (Ext R J are two proper ideals of R, such that J ⊆ I , then J is said to be a reduction of I if I n+1 = J I n for all n 0. We set red I (J ) = min{n ∈ N | I n+1 = J I n }
Communicated by Rahim Zaare-Nahandi.
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Mehran Rahimi [email protected] Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran
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Bulletin of the Iranian Mathematical Society
and call it as the reduction number of I with respect to J . The symbol red (I ) = min{ red I (J ) | J is a reduction of I } stands for reduction number of I . The aim of this paper is to study Cohen–Macaulay local rings possessing canonical ideals. Over a Cohen–Macaulay local ring (R, m, k) of dimension d, a maximal Cohen–Macaulay module ω is called a canonical module of R if dimk Ext iR (k, ω) = δid . An ideal of R, isomorphic to a canonical module, is called a canonical ideal of R. A local ring admits a canonical module if and only if it is a homomorphic image of a Gorenstein ring. Such a ring R admits a canonical ideal if and only if Rp is Gorenstein for all associated prime ideals p of R. (For more informations on canonical modules, see [17] and [2, Section 3.3]). In Sect. 2, for a fixed R, we denote by C R , the set of all canonical ideals of R and investigate it in details. Here is a result of this kind. Result 1.1 (Theorem 2.5) Let I and J belong to C R . (a) If I ⊂ J , then I = x J for some x ∈ m where x R = (I : J ). (b) If x ∈ I is a regular element, then x J = y I for some y ∈ J . (c) Let dim R = 1. If red (m) = t, then mt+1 does not contain any element of Max (C R ), where red (m) denotes the reduction nu
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