On derivations and commutativity of prime rings with involution

Abstract

In [6], Bell and Daif proved that if R is a prime ring admitting a nonzero derivation such that d(xy)=d(yx) for all x,y ? R, then R is commutative. The objective of this paper is to examine similar problems when the ring R is equipped with involution. It is shown that if a prime ring R with involution * of a characteristic different from 2 admits a nonzero derivation d such that d(XX * )=d(x * x) for all x ? R and S(R) ? Z(R) ? (0), then R is commutative. Moreover, some related results have also been discussed. © 2016 by De Gruyter 2016.