Semicommutativity of the rings relative to prime radical

Abstract

In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called P-semicommutative. We prove that a ring R is P-semicommutative if and only if R[x] is P-semicommutative if and only if R[x, x(-1)] is P-semicommutative. Also, if R[[x]] is P-semicommutative, then R is P-semicommutative. The converse holds provided that P(R) is nilpotent and R is power serieswise Armendariz. For each positive integer n, R is P-semicommutative if and only if T-n (R) is P-semicommutative. For a ring R of bounded index 2 and a central nilpotent element s, R is P-semicommutative if and only if K-s (R) is P-semicommutative. If T is the ring of a Morita context (A, B, M, N,Psi, phi with zero pairings, then T is P-semicommutative if and only if A and B are P-semicommutative. Many classes of such rings are constructed as well. We also show that the notions of clean rings and exchange rings coincide for P-semicommutative rings.