Noetherian semigroup algebras
type: Item
instance of: scholarly article
title: Noetherian semigroup algebras
zbMATH Open document ID: 0937.16043
author: Eric Jespers
author: Jan Okniński
published in: Journal of Algebra
publication date: 1999-12-20T00:00:00Z
review text: Let \(S\) be a semigroup and let \(K\) be a field. The authors are interested in conditions under which the semigroup algebra \(K[S]\) is right or left Noetherian. In particular, they investigate cases when these two concepts coincide, when Noetherian property of \(K[S]\) implies \(S\) is finitely generated, and when \(S\) satisfies the ascending chain conditions on one-sided ideals. Main results sound as follows. Suppose that \(S\) is a submonoid of a finitely generated nilpotent-by-finite group; then the following three conditions are equivalent: \(K[S]\) is right Noetherian, \(K[S]\) is left Noetherian, \(S\) satisfies the ascending chain conditions on one-sided ideals (Corollary 1.6). Suppose that either \(S\) is a submonoid of a finitely generated nilpotent-by-finite group, or \(K[S]\) has finite Gelfand-Kirillov dimension, or else \(S\) is Malcev nilpotent; then \(S\) is finitely generated (Theorem 2.2). A description of prime Goldie semigroup algebras is also given, a generalization of Zelmanov's well known results on prime PI-semigroup algebras.
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16R99
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 20F18
zbMATH DE Number: 1348171
zbMATH Keywords: semigroup algebras
zbMATH Keywords: ascending chain conditions
zbMATH Keywords: finitely generated nilpotent-by-finite groups
zbMATH Keywords: Gelfand-Kirillov dimensions
zbMATH Keywords: Malcev nilpotent semigroups
zbMATH Keywords: prime Goldie semigroup algebras
reviewed by: Josif S. Ponizovskii
MaRDI profile type: Publication
full work available at URL: jabr.1999.7870
OpenAlex ID: W1967547293
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DOI: 10.1006/JABR.1999.7870
description: scientific article; zbMATH DE number 1348171