Semigroup algebras of linear semigroups
type: Item
instance of: scholarly article
title: Semigroup algebras of linear semigroups
zbMATH Open document ID: 0770.20027
DOI: 10.1016/0021-8693(92)90116-4
published in: Journal of Algebra
publication date: 1993-05-16T00:00:00Z
review text: The authors state that in this paper they begin a systematic study of the semigroup algebras of linear semigroups, i.e. the subsemigroups of the multiplicative monoids \({\mathcal M}_ n(k)\) of all \(n\times n\)-matrices over a field \(k\), which is supposed to be algebraically closed. \({\mathcal M}_ n(k)\) is considered with respect to its Zariski topology. If \(F\) is a field, \(F[S]\) denotes the semigroup algebra of a semigroup \(S\). First, the connections between \(F[S]\) and \(F[\overline S]\), where \(S\subset {\mathcal M}_ m(k)\) and \(\overline S\) is its (Zariski) closure, are established, and then only the case is considered, when \(S =\overline S\) is a connected monoid. [See \textit{M. S. Putcha}, ``Linear Algebraic Monoids'' (Lond. Math. Soc. Lect. Note Ser. 133, 1988; Zbl 0647.20066).] It is shown, that if \(F\) is any field of characteristic zero and \(M\) is a closed connected monoid, then the Jacobson radical \(J(F[M])\) is the kernel of the homomorphism \(F[M] \to F[M/\sim]\) for some congruence \(\sim\) on \(M\). If \(M\) is a connected monoid with zero, and \(F\) is any field, then \(F[M]\) is semisimple iff \(M\) is regular and the characteristic of \(F\) is either zero or else equals the characteristic of \(K\). There are more interesting results. Finally it is proved that if \(R\) is a unitary algebra over a field \(k\), and \(R^*\) denotes the multiplicative semigroup of \(R\), then for any field \(F\) of characteristic zero, \(F[R^*]\) is semisimple iff \(R\) is semiprime.
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 165839
zbMATH Keywords: semigroups of matrices
zbMATH Keywords: semigroup algebras of linear semigroups
zbMATH Keywords: multiplicative monoids
zbMATH Keywords: connected monoid
zbMATH Keywords: Jacobson radical
zbMATH Keywords: semisimple
zbMATH Keywords: semiprime
author: Jan Okniński
author: Mohan S. Putcha
reviewed by: Grigori I. Zhitomirski
MaRDI profile type: Publication
full work available at URL: 0021-8693(92)90116-4
OpenAlex ID: W2001721372
cites work: Some theorems on actions of algebraic groups
cites work: Q4126563
cites work: Q3929934
cites work: On commutative semigroup algebras
cites work: On permutative semigroup algebras
cites work: Strongly π-Regular Matrix Semigroups
cites work: On cancellative semigroup rings
cites work: NOETHERIAN SEMIGROUP ALGEBRAS
cites work: Complex Representations of Matrix Semigroups
cites work: Pi Semigroup Algebras of Linear Semigroups
cites work: On the semisimplicity of group rings of some locally finite groups
cites work: Observations on group rings
cites work: Matrix Semigroups
cites work: Reductive groups and regular semigroups
cites work: The system of idempotents and the lattice of \({\mathcal J}\)-classes of reductive algebraic monoids
cites work: Reductive monoids are von Neumann regular
cites work: Classification of Semisimple Algebraic Monoids
cites work: Completely regular algebraic monoids
cites work: Q4135685
cites work: Linear algebraic groups
cites work: Q5676405
cites work: On group rings of linear groups
Recommended article: Q3972281
Recommended article: The kernel of a linear algebraic semigroup
Recommended article: Subspace semigroups
Recommended article: Radicals of algebras graded by cancellative linear semigroups
Recommended article: Q4370156
Recommended article: Q3682679
Recommended article: Q4220583
Recommended article: Q4040248
Recommended article: Q3973321
Recommended article: On semigroup algebras satisfying polynomial identities
description: scientific article; zbMATH DE number 165839