Local rings of minimal length
type: Item
instance of: scholarly article
title: Local rings of minimal length
zbMATH Open document ID: 1162.13002
author: Valentina Barucci
published in: Journal of Pure and Applied Algebra
publication date: 2009-03-30T00:00:00Z
review text: Let \((R,\mathfrak m)\) denote a local ring. A regular fractional ideal \(\omega\) of \(R\) is called an \(\mathfrak m\)-canonical ideal provided \(\omega : (\omega : I) = I\) for any regular fractionary ideal \(I\) of \(R.\) Assume that \(R\) possesses an \(\mathfrak m\)-canonical ideal \(\omega,\) and let \(t\) denote its minimal number of generators. Then \(l_R(\omega/R) \geq t-1,\) and \(R\) is called of minimal length if equality holds. The author considered the following conditions: (1) \(R\) is of minimal length. (2) Each regular ideal of \((\mathfrak m : \mathfrak m)\) is divisorial as ideal of \(R.\) (3) \(\mathfrak m\) is an \(\mathfrak m\)-canonical ideal of \((\mathfrak m : \mathfrak m).\) Then it is shown that (1)\(\Longrightarrow\)(2)\(\Longleftrightarrow\)(3). Moreover, all the conditions are equivalent for \(R\) a one-dimensional analytically unramified and residually rational ring. As illustrations of the results several examples are considered.
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 5534968
zbMATH Keywords: canonical ideal
zbMATH Keywords: fractionary ideal
zbMATH Keywords: minimal length
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.11.007
OpenAlex ID: W2134276214
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DOI: 10.1016/J.JPAA.2008.11.007
description: scientific article; zbMATH DE number 5534968