On algebraically compact semigroup rings
type: Item
instance of: scholarly article
title: On algebraically compact semigroup rings
zbMATH Open document ID: 0669.20051
DOI: 10.1007/BF02573299
author: A. V. Klyushin
author: I. B. Kozhukhov
published in: Semigroup Forum
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134988
review text: It is known that a group ring R[G] is algebraically compact if and only if G is a finite group and the ring R is algebraically compact [\textit{W. Zimmermann}, J. Pure Appl. Algebra 23, 319-328 (1982; Zbl 0474.16016)]. However, there exist (right) algebraically compact semigroup rings R[S] of infinite semigroups. The authors determine several necessary conditions on the structure of semigroups S of this type, extending the results obtained by the reviewer for the class of right self-injective semigroup rings [Arch. Math. 43, 407-411 (1984; Zbl 0534.20038)]. While these conditions are not sufficient in general, they entirely characterize algebraically compact semigroup rings of completely 0-simple semigroups. We note that it is not known whether S must be finite whenever R[S] is right self-injective. It was proved by the reviewer that, if R is a field, then R[S] is right perfect in this case. This, together with Osofsky's results, implies that S must be finite whenever R[S] is both left and right self-injective. On the other hand, in a forthcoming paper the authors prove that R[S] is left linearly compact (in the discrete topology) if and only if so is R and S is a finite semigroup.
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16W80
zbMATH DE Number: 4095785
zbMATH Keywords: group ring
zbMATH Keywords: algebraically compact semigroup rings
zbMATH Keywords: right self-injective semigroup rings
zbMATH Keywords: completely 0-simple semigroups
zbMATH Keywords: left linearly compact
reviewed by: Jan Okniński
MaRDI profile type: Publication
cites work: Q3848243
cites work: On the Group Ring
cites work: Q5682108
cites work: Linearly compact semigroup rings
cites work: Q5619196
cites work: Purity and algebraic compactness for modules
cites work: Rein injektive direkte summen von moduln
cites work: (Sigma-) algebraic compactness of rings
OpenAlex ID: W2090088971
description: scientific article; zbMATH DE number 4095785