Faithful linear representations of bands.
type: Item
instance of: scholarly article
title: Faithful linear representations of bands.
zbMATH Open document ID: 1178.20054
DOI: 10.5565/PUBLMAT_53109_06
author: Ferran Cedó
author: Jan Okniński
published in: Publicacions Matemàtiques
publication date: 2009-01-29T00:00:00Z
review text: An attractive problem in ring theory is the embeddability of a ring \(R\) into the full or triangular matrix ring over a commutative ring. In this paper the authors investigate such questions for semigroup algebras \(K[S]\) of a band \(S\) and a field \(K\). Recall that a band is a semigroup consisting of idempotents. Such a semigroup is a semilattice \(S=\bigcup_{\alpha\in\Gamma}S_\alpha\) of rectangular bands \(S_\alpha\). These are called the components of \(S\). In the first part of the paper it is proved that if a band \(S\) has finitely many components then \(K[S]\) can be embedded in upper triangular matrices over a commutative \(K\)-algebra. An explicit embedding is given in case \(S\) has two components. In the second part, the proof of a theorem of \textit{A. I. Mal'tsev} [Mat. Sb., N. Ser. 13(55), 263-286 (1943; Zbl 0060.07804), Theorem 10] on embeddability of algebras into matrix algebras over a field is corrected. Furthermore, it is proved that if \(S=F\cup E\) is a band with two components \(E\) and \(F\), such that \(F\) is an ideal of \(S\) and \(E\) is finite, then \(S\) is a linear semigroup. For some special classes of bands satisfying finiteness conditions on annihilators, some explicit embeddings of the respective semigroups, and their semigroup algebras, into triangular matrices over fields are given.
reviewed by: Eric Jespers
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M30
zbMATH DE Number: 5500105
zbMATH Keywords: linear bands
zbMATH Keywords: semigroup algebras
zbMATH Keywords: triangular matrices
zbMATH Keywords: annihilators
zbMATH Keywords: PI rings
zbMATH Keywords: normal bands
zbMATH Keywords: matrix representations
zbMATH Keywords: algebras of matrices
zbMATH Keywords: semilattices
zbMATH Keywords: rectangular bands
MaRDI profile type: Publication
full work available at URL: publmat 53109 06
OpenAlex ID: W2093274477
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description: scientific article; zbMATH DE number 5500105