\(RD\)-formulas and \(W\)-rings
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.
scientific article; zbMATH DE number 567854