Identities of orthodox semigroup rings
type: Item
instance of: scholarly article
title: Identities of orthodox semigroup rings
zbMATH Open document ID: 0815.20065
DOI: 10.1007/BF02573486
author: Guangtian Song
published in: Semigroup Forum
publication date: 1995-06-30T00:00:00Z
full work available at URL: https://eudml.org/doc/135349
review text: Let \(R\) be a ring with identity, let \(S\) be a semigroup, and let \(T\) be the subsemigroup of \(S\) generated by all idempotents of \(S\). The semigroup ring of \(S\) over \(R\) is denoted by \(R[S]\). The author is interested in the two following problems. Problem 1: when is \(R[S]\) a ring with identity? Problem 2: suppose that \(R[S]\) is a ring with identity, and \(S\) is regular; is it true that then \(R[T]\) is also a ring with identity? Problem 1 was raised by the reviewer [in Semigroup Forum 36, 1-46 (1987; Zbl 0629.20039)], Problem 2 has been proposed by \textit{F. Li} [ibid. 46, 27-31 (1993; Zbl 0787.16024)]. The most interesting result of the paper is the affirmative answer to problem 2 for \(S\) orthodox (Theorem 2.1). The paper contains several results concerning Problem 1 for orthodox semigroups. Denote by \(Z_ R\) the subring of \(R\) generated by the identity of \(R\). The author shows that Problem 1 for \(S\) orthodox may be reduced to the same problem for the semigroup ring \(Z_ R[B]\) for a specially chosen finite band \(B\) from \(S\). In view of this result, the author investigates in the last section of the paper Problem 1 for semigroup rings of finite bands over \(Z_ R\).
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 20M19
Mathematics Subject Classification ID: 20M17
zbMATH DE Number: 638355
zbMATH Keywords: regular semigroups
zbMATH Keywords: ring with identity
zbMATH Keywords: idempotents
zbMATH Keywords: orthodox semigroups
zbMATH Keywords: semigroup rings
zbMATH Keywords: finite bands
reviewed by: Josif S. Ponizovskii
MaRDI profile type: Publication
cites work: Q3848243
cites work: Q4126563
cites work: The existence of identity of orthodox semigroup rings
cites work: Q4148155
cites work: Semigroup rings
cites work: Self-Injective Semigroup Rings for Finite Inverse Semigroups
description: scientific article; zbMATH DE number 638355