Some generalizations in certain classes of rings with involution - doi: 10.5269/bspm.v29i1.11384

Abstract

Let R be a 2-torsion free sigma-prime ring with an involution sigma, I a nonzero sigma-ideal of R. In this paper we explore the commutativity of R satisfying any one of the properties: (i) d(x) oF(y) = 0 for all x, y ∈ I. (ii) [d(x), F(y)] = 0 for all x, y ∈ I. (iii) d(x) o F(y) = x o y for all x, y ∈ I. (iv) d(x)F(y) − xy ∈ Z(R) for
all x, y ∈ I. We also discuss (alpha,beta )-derivations of sigma-prime rings and prove that if G is an (alpha,beta)-derivation which acts as a homomorphism or as an anti-homomorphism on I, then G = 0 or G = beta on I.