A decomposition theorem for \(\wp^*\)-semisimple rings.
type: Item
instance of: scholarly article
title: A decomposition theorem for \(\wp^*\)-semisimple rings.
zbMATH Open document ID: 1048.16010
DOI: 10.1016/S0022-4049(03)00127-0
author: Hai Quang Dinh
author: Dinh Van Huynh
published in: Journal of Pure and Applied Algebra
publication date: 2004-01-26T00:00:00Z
review text: An associative ring \(R\) with identity is defined to be right (left) \(\wp^*\)-semisimple if every right (left) \(R\)-module is the direct sum of a projective module and a quasi-continuous module. As is pointed out in the paper, right \(\wp^*\)-semisimple rings are right Artinian, but need not be left \(\wp^*\)-semisimple. In fact, right \(\wp^*\)-semisimple rings are precisely those rings \(R\) over which every right \(R\)-module is the direct sum of a projective module and a quasi-injective module. The present paper discusses rings which are both right and left \(\wp^*\)-semisimple. The main result of the paper is a structure theorem of a right and left \(\wp^*\)-semisimple ring \(R\), representing \(R\) as a ring-theoretic direct sum of five rings satisfying some specific conditions. As a consequence, \(R\) is \(\wp^*\)-semisimple if and only if there is a ring-direct decomposition \(R=D\oplus E\), where every right (and left) \(D\)-module is the direct sum of a projective module and a semisimple module, and every right (and left) \(E\)-module is the direct sum of a projective module and an injective module.
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 2032647
zbMATH Keywords: \(\wp^*\)-semisimple rings
zbMATH Keywords: quasi-continuous modules
zbMATH Keywords: quasi-injective modules
zbMATH Keywords: direct sums
zbMATH Keywords: right Artinian rings
zbMATH Keywords: projective modules
zbMATH Keywords: semisimple modules
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2032647