Semigroup algebras that are principal ideal rings
type: Item
instance of: scholarly article
title: Semigroup algebras that are principal ideal rings
zbMATH Open document ID: 0857.16028
published in: Journal of Algebra
publication date: 1997-01-14T00:00:00Z
review text: Let \(K\) be a field and let \(S\) be a semigroup. If the semigroup ring \(K[S]\) is a principal left ideal ring, then \(K[S]\) satisfies a polynomial identity, and hence \(S\) is finitely generated, \(K[S]\) embeds into a matrix ring \(M_n(F)\) over a field extension \(F\) of \(K\), and the Gelfand-Kirillov dimension of \(K[S]\) equals the classical Krull dimension of \(K[S]\). The technical structure of the finite semigroups \(S\) for which \(K[S]\) is a principal ideal ring is completely characterized. If \(K[S]\) is a semiprime principal left ideal ring, then \(K[S]\) is also a principal right ideal ring. Every prime contracted semigroup algebra \(K_0[S]\) must be of the form \(M_n(K)\), \(M_n(K[X])\), or \(M_n(K[X,X^{-1}])\), and the structure of \(S\) is completely determined.
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 20M35
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16U30
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16U80
zbMATH DE Number: 923706
zbMATH Keywords: Munn algebras
zbMATH Keywords: semigroup rings
zbMATH Keywords: principal left ideal rings
zbMATH Keywords: polynomial identities
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: classical Krull dimension
zbMATH Keywords: finite semigroups
zbMATH Keywords: semiprime principal left ideal rings
zbMATH Keywords: principal right ideal rings
zbMATH Keywords: prime contracted semigroup algebras
author: Eric Jespers
author: Jan Okniński
reviewed by: Mark L. Teply
MaRDI profile type: Publication
full work available at URL: jabr.1996.0241
OpenAlex ID: W2072780423
DOI: 10.1006/JABR.1996.0241
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description: scientific article; zbMATH DE number 923706