Principally injective rings
type: Item
instance of: scholarly article
title: Principally injective rings
zbMATH Open document ID: 0839.16004
published in: Journal of Algebra
publication date: 1996-06-19T00:00:00Z
review text: A ring \(R\) is called right principally injective if every \(R\)-homomorphism from a principal right ideal of \(R\) is a left multiplication by an element of \(R\). Semiperfect, right principally injective rings with essential right socles, which are a natural generalization of the pseudo-Frobenius rings, are studied. In fact for such a ring the following interesting facts are shown: (1) that the right socle equals the left socle, and it is essential on both sides and is finitely generated on the left; (2) that the two singular ideals coincide; and (3) that such a ring has a Nakayama permutation of its basic idempotents. Furthermore, principally injective group rings are investigated. Indeed it is proved that if the group ring \(RG\) is right principally injective, then \(R\) is right principally injective and \(G\) is locally finite; and that if \(R\) is right self-injective and \(G\) is locally finite, then the group ring \(RG\) is right principally injective.
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 783854
zbMATH Keywords: locally finite groups
zbMATH Keywords: principal right ideals
zbMATH Keywords: left multiplication
zbMATH Keywords: right principally injective rings
zbMATH Keywords: right socles
zbMATH Keywords: pseudo-Frobenius rings
zbMATH Keywords: singular ideals
zbMATH Keywords: Nakayama permutation
zbMATH Keywords: basic idempotents
zbMATH Keywords: principally injective group rings
Wikidata QID: Q122961336
author: W. Keith Nicholson
author: Mohamed F. Yousif
reviewed by: Jae Keol Park
MaRDI profile type: Publication
full work available at URL: jabr.1995.1117
OpenAlex ID: W2065471045
DOI: 10.1006/JABR.1995.1117
description: scientific article; zbMATH DE number 783854