On derivations and commutativity of prime rings with involution

Abstract

In [6], Bell and Daif proved that if is a prime ring admitting a nonzero derivation such that ( ) = ( ) for all , ∈ , then is commutative. The objective of this paper is to examine similar problems when the ring is equipped with involution. It is shown that if a prime ring with involution * of a characteristic di erent from 2 admits a nonzero derivation such that ( * ) = ( * ) for all ∈ and ( ) ∩ ( ) ̸ = (0), then is commutative. Moreover, some related results have also been discussed.

On $^*$-differential identities in prime rings with involution

“…In this paper, our intent is to continue to investigate and discuss the commutativity of prime rings with involution ' * " satisfying certain *differential identities. In fact, our results generalize and unify several well known and classical theorems proved in [4], [12], and [16].…”

On generalized derivations and commutativity of prime rings with involution

“…We facilitate our discussion with the following theorem which generalized many know results proved in [4], [5] and [13]. Precisely, first we give a brief proof of We now prove the next theorem in same domain.…”

“…In this paper, our aim is to continue this line of investigation and discuss the commutativity of prime rings with involution involving generalized derivations. In particular, we extended some results proved in [2], [4], [5], [8] and [13] for derivations to generalized derivations.…”

Classification of Some Special Generalized Derivations

“…Moreover, many of obtained results extend other ones previously proven just for the action of the considered mapping on the whole ring. In this direction, the recent literature contains numerous results on commutativity in prime and semi-prime rings admitting suitably constrained derivations and generalized derivations, and several authors have improved these results by considering rings with involution (for example, see [1], [2], [13]).…”