\(RD\)-formulas and \(W\)-rings
type: Item
instance of: scholarly article
title: \(RD\)-formulas and \(W\)-rings
zbMATH Open document ID: 0809.16006
author: G. E. Puninskij
published in: Mathematical Notes
publication date: 1994-07-12T00:00:00Z
review text: A ring \(R\) is called a right (left) \(W\)-ring if any finitely presented right (left) \(R\)-module is a direct summand of a direct sum of modules of the form \(R/rR\) (\(R/Rr\)), \(r \in R\). A ring \(R\) is called a \(W\)-ring if \(R\) is a right \(W\)-ring and a left \(W\)-ring. In this paper, a matrix characterization of right \(W\)-rings is considered, some results about \(W\)-rings are established, many examples of \(W\)-rings are given. In particular, the author studies conditions under which local \(W\)-rings are uniserial. For example, (1) A local domain \(R\) is a \(W\)-ring if and only if \(R\) is uniserial; (2) A local ring \(R\) with \(J^ \infty = \{0\}\) is a right \(W\)-ring if and only if \(R\) is uniserial, where \(J\) is the Jacobson radical of \(R\); (3) A local left or right Noetherian or left or right perfect ring \(R\) is a \(W\)-ring if and only if \(R\) is uniserial.
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 16B50
zbMATH DE Number: 567854
zbMATH Keywords: FP-injective module
zbMATH Keywords: direct sum of modules
zbMATH Keywords: right \(W\)-rings
zbMATH Keywords: local \(W\)- rings
zbMATH Keywords: uniserial
zbMATH Keywords: local domain
zbMATH Keywords: Jacobson radical
zbMATH Keywords: perfect ring
reviewed by: Jule Zhang
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 567854