Local multiplications on algebras
type: Item
instance of: scholarly article
title: Local multiplications on algebras
zbMATH Open document ID: 0868.16026
DOI: 10.1016/S0022-4049(96)00013-8
author: Jeanne Wald Kerr
author: Don Hadwin
published in: Journal of Pure and Applied Algebra
publication date: 1997-08-14T00:00:00Z
review text: Let \(R\) be a commutative ring with identity and let \(A\) be an algebra with identity over \(R\). It is said that an \(R\)-endomorphism \(\varphi\) on \(A\) is LIP (left-ideal-preserving) if \(\varphi(M)\subset M\) for every left ideal \(M\) of \(A\). The algebra \(A\) is said to be SLIP over \(R\) if every LIP \(R\)-endomorphism is a left multiplication by an element of \(A\). The main results proved by the authors are: 1) Suppose \(\{F_i:i\in I\}\) is a collection of two-sided ideals in \(A\) such that \(A/F_i\) is SLIP over \(R\) for every \(i\in I\) and \(\bigcap F_i=\{0\}\), then \(A\) is SLIP over \(R\); 2) Suppose \(R\) is an integral domain. The following are equivalent: (i) \(R\) is infinite. (ii) \(R[S]\) is SLIP over \(R\) for every nonempty set \(S\). (iii) There is a nonempty set \(S\) such that \(R[S]\) is SLIP over \(R\). 3) Suppose \(R\) is an integral domain containing the field of rational numbers \(\mathbb{Q}\) and that, for every positive integer \(n\), \(R\) contains \(n\) distinct \(n\)th roots of unity. Suppose \(G\) is a group that is a subdirect product of torsion groups. Then the group ring \(R[G]\) is SLIP over \(R\).
reviewed by: Iuliu Crivei
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 987941
zbMATH Keywords: left ideal preserving endomorphisms
zbMATH Keywords: LIP endomorphisms
zbMATH Keywords: left multiplications
zbMATH Keywords: integral domains
zbMATH Keywords: subdirect products of torsion groups
zbMATH Keywords: group rings
Wikidata QID: Q114122621
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 987941