Rigid, quasi-rigid and matrix rings with (σ, 0)-multiplication

Abstract

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