On Generalized Derivations and Commutativity of Prime Rings with Involution

Let R be a ring with involution ′∗′. A map δ of the ring R into itself is called a derivation if δ(xy) = δ(x)y + xδ(y) for all x, y ∈ R. An additive map F : R → R is called a generalized derivation on R if F(xy) = F(x)y + xδ(y) for all x, y ∈ R, Permanent address: Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh202002, India 292 Shakir Ali and Husain Alhazmi where δ is a derivation of R. In [1, Theorem 2.2], we proved that if a prime ring R with involution′∗′ of the second kind and char(R) 6= 2 admits a nonzero generalized derivation F such that F([x, x∗]) = 0 for all x ∈ R, then R is commutative. In fact, the proof of above mentioned result and [1, Theorem 2.5] are complicated and technical. The aim of this manuscript is to give a brief and elegant proofs of these results. As an application, and apart from proving the other results, many known theorems can be either generalized or deduced. Mathematics Subject Classification: 16N60; 16W10; 16W25