Rigid, quasi-rigid and matrix rings with (?,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 S<inf>n,m</inf>(R) is a ring with (?, 0)-multiplication. © 2014 Elsevier B.V., All rights reserved.