One-dimensional almost Gorenstein rings
type: Item
instance of: scholarly article
title: One-dimensional almost Gorenstein rings
zbMATH Open document ID: 0874.13018
author: Valentina Barucci
author: Ralf Fröberg
published in: Journal of Algebra
publication date: 1997-10-26T00:00:00Z
review text: Let \(A\) be an one-dimensional local Cohen-Macaulay ring with finite closure \(\bar A\). It is well-known that \(\ell (\bar A/A) \geq \ell (A/C) + \text{type}(A)-1\), where \(C = A:\bar A\). If \(A\) is a Gorenstein ring, then \(\text{type}(A) = 1\) and \(\ell(\bar A/A) = \ell(A/C)\). For this reason the authors call \(A\) an almost Gorenstein ring if \(\ell(\bar A/A) = \ell(A/C) + \text{type}(A) - 1\). They study this class of local rings under the assumption that \(A\) has a canonical ideal \(K\) such that \(A \subseteq K \subseteq \bar A\). If \(A\) is analytically irreducible and residual, one can assign with \(A\) a numerical semigroup \(v(A)\). \textit{E. Kunz} [Proc. Am. Math. Soc. 25, 748-751 (1970; Zbl 0197.31401)] proved that \(A\) is Gorenstein if and only if \(v(A)\) is symmetric. Inspired of this result, the authors show that almost Gorenstein rings and some of their properties can be similarly characterized in terms of \(v(A)\).
reviewed by: Trung Ngo Viet
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 1013931
zbMATH Keywords: almost Gorenstein ring
zbMATH Keywords: Kunz ring
zbMATH Keywords: analytically irreducible ring
MaRDI profile type: Publication
full work available at URL: jabr.1996.6837
OpenAlex ID: W2020485388
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DOI: 10.1006/JABR.1996.6837
description: scientific article; zbMATH DE number 1013931