Rigid, quasi-rigid and matrix rings with \((\overline{\sigma},0)\)multiplication

Abstract

Let \(R\) be a ring with an endomorphism \(\sigma\). We introduce \((\overline{\sigma}, 0)\)-multiplication which is a generalization of the simple \( 0\)- multiplication. It is proved that for arbitrary positive integers \(m\leq n\) and \(n\geq 2\), \(R[x; \sigma]\) is a reduced ring if and only if \(S_{n, m}(R)\) is a ring with \((\overline{\sigma},0)\)-multiplication.