When is the Jacobson radical of a semigroup ring of a commutative semigroup homogeneous?
type: Item
instance of: scholarly article
title: When is the Jacobson radical of a semigroup ring of a commutative semigroup homogeneous?
zbMATH Open document ID: 0619.20045
DOI: 10.1016/0021-8693(87)90155-4
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let S be a commutative semigroup and let R be an associative ring such that \(R\cap J(R[x])=R\cap J(R[x_ 1,...,x_ n])\) for any \(n>1\) where J denotes the Jacobson radical. Based on the author's description of J(R[S]) [ibid. 109, 266-280 (1987; Zbl 0617.20040)], necessary and sufficient conditions for J(R[S]) to be S-homogeneous are determined. In particular, it is shown that in this case \(J(R[S])=(R\cap J(R[S]))[S]+J(R)[P]\) where P denotes the largest ideal of S consisting of periodic elements. While in general the result obtained does not provide a complete intrinsic characterization of the homogeneity of J(R[S]) in terms of S, such a result is obtained for some special classes of semigroups and the difficulties when dealing with the general case are illustrated.
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16Nxx
Mathematics Subject Classification ID: 16W99
zbMATH DE Number: 4004487
zbMATH Keywords: commutative semigroup
zbMATH Keywords: Jacobson radical
zbMATH Keywords: homogeneity
author: Eric Jespers
reviewed by: Jan Okniński
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90155-4
OpenAlex ID: W2003113366
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description: scientific article; zbMATH DE number 4004487