Rigid, quasi-rigid and matrix rings with ((?)over-bar, 0)-multiplication

Abstract

Let R be a ring with an endomorphism sigma. We introduce ((sigma) over bar, 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; sigma] is a reduced ring if and only if S-n,S-m (R) is a ring with ((sigma) over bar, 0)-multiplication.