A construction of Gorenstein rings
type: Item
instance of: scholarly article
title: A construction of Gorenstein rings
zbMATH Open document ID: 1120.13022
author: M. D'Anna
published in: Journal of Algebra
publication date: 2007-01-11T00:00:00Z
review text: Let \(R\) be a local Cohen-Macaulay ring and \(M\) be an \(R\)-module. Recall that the trivial extension, \(R\ltimes M\), of \(R\) by \(M\), is the \(R\)-algebra \(R\oplus M\) whose multiplication makes \((1,M)\) an ideal which squares to zero. A theorem of \textit{I. Reiten} [Proc. Am. Math. Soc. 32, 417--420 (1972; Zbl 0235.13016)], says that \(R\ltimes M\) is a Gorenstein ring if and only if \(M\) is a canonical module of \(R\). The present paper considers a similar construction \(R\bowtie I\), where \(I\) is an ideal of \(R\). (The ring \(R\bowtie I\) is reduced whenever \(R\) is reduced, so \(\bowtie\) is definitely different than \(\ltimes\).) The result in the present paper is parallel to Reiten's result: the ring \(R\bowtie I\) is Gorenstein if and only if \(I\) is isomorphic to a canonical module of \(R\). The construction is applied to rings of the form \(R=kt_1,\dots,t_n/(P_1\cap \dots \cap P_h)\), where each \(P_i\) is a prime ideal of height \(n-1\). Such rings are called algebroid curves with \(h\) branches. It is shown that \(R\bowtie I\) is an algebroid curve with \(2h\) branches.
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 5115353
zbMATH Keywords: algebroid curve
zbMATH Keywords: trivial extension
reviewed by: Andrew Kustin
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2005.12.023
OpenAlex ID: W2116495070
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DOI: 10.1016/J.JALGEBRA.2005.12.023
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description: scientific article; zbMATH DE number 5115353