On the semiprimitivity of inverse semigroup algebras and on theorems by Domanov and Munn
type: Item
instance of: scholarly article
title: On the semiprimitivity of inverse semigroup algebras and on theorems by Domanov and Munn
zbMATH Open document ID: 0683.20053
DOI: 10.1007/BF02573265
published in: Semigroup Forum
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/135017
review text: It was shown by \textit{O. I. Domanov} [Mat. Issled. 38, 123-131 (1976; Zbl 0411.20043)] that the semigroup ring K[S] of an inverse semigroup S over a field K is semiprimitive whenever all group rings K[G], G a subgroup of S, are semiprimitive. The converse does not hold in general, but was shown to be true if the lattice E(S) of idempotents of S is a, so called, pseudofinite lattice [\textit{W. D. Munn}, Proc. R. Soc. Edinb., Sect. A 107, 175-196 (1987; Zbl 0627.20041)]. In the paper under review, for a given non-pseudofinite lattice E, the author constructs an inverse semigroup S such that \(E(S)=E\), K[S] is semiprimitive for some field K, and there exists a subgroup G of S such that K[G] is not semiprimitive. This shows that the converse of Domanov's theorem holds for all S with a given lattice of idempotents E, and for all fields K, exactly when E is pseudofinite.
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 4118617
zbMATH Keywords: semiprimitive ring
zbMATH Keywords: semigroup ring
zbMATH Keywords: inverse semigroup
zbMATH Keywords: group rings
zbMATH Keywords: idempotents
zbMATH Keywords: pseudofinite lattice
zbMATH Keywords: Domanov's theorem
author: Josif S. Ponizovskii
reviewed by: Jan Okniński
MaRDI profile type: Publication
cites work: Q4198943
cites work: A class of contracted inverse semigroup rings
cites work: Q4148155
cites work: An example of a semiprimitive semigroup algebra
cites work: On semigroup rings
cites work: On Semisimple Semigroup Rings
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description: scientific article; zbMATH DE number 4118617