On permutative semigroup algebras
type: Item
instance of: scholarly article
title: On permutative semigroup algebras
zbMATH Open document ID: 0654.20082
DOI: 10.1007/BF01229980
author: Thomas E. Nordahl
published in: Algebra Universalis
publication date: 1988-01-01T00:00:00Z
review text: Let F be a field and let S be a permutative semigroup, that is a semigroup with a non-trivial permutation identity \(x_ 1x_ 2...x_ n=x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)}\). It is shown that the Jacobson radical of the semigroup algebra F[S] is a nil ideal and it coincides with the subspace of F[S] spanned by the set \(\{\) s-t\(|\) (s,t)\(\in \nu \}\) for a congruence \(\nu\) on S. Here \(\nu\) is the least separative congruence on S if \(ch(F)=0\), and ``the least p-separative'' congruence if \(ch(F)=p>0\). The second main result establishes a criterion for the semisimplicity of a PI semigroup algebra of an arbitrary separative semigroup. These results extend the description of the radical of commutative semigroup algebras due to \textit{W. D. Munn} [Math. Proc. Camb. Philos. Soc. 93, 237-246 (1983; Zbl 0528.20053)], and the results of the reviewer [ibid. 99, 45-50 (1986; Zbl 0583.20052)] on the radical of semigroup algebras satisfying polynomial identities.
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16Nxx
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 16W99
zbMATH DE Number: 4067250
zbMATH Keywords: permutative semigroup
zbMATH Keywords: permutation identity
zbMATH Keywords: Jacobson radical
zbMATH Keywords: semigroup algebra
zbMATH Keywords: nil ideal
zbMATH Keywords: separative congruence
zbMATH Keywords: semisimplicity
zbMATH Keywords: PI semigroup algebra
zbMATH Keywords: separative semigroup
zbMATH Keywords: semigroup algebras satisfying polynomial identities
reviewed by: Jan Okniński
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01229980
OpenAlex ID: W1993372935
description: scientific article; zbMATH DE number 4067250