Abstract
Herstein proved that a prime ring R of is commutative if there is a nonzero derivation d of R such that
for all
The aim of this paper is to prove the
-version of Herstein’s result with a pair of derivations on prime ideals of a ring with involution. Precisely, we prove the following result: let R be a ring with involution
of the second kind, P a prime ideal of R such that
and
If d1 and d2 are derivations of R satisfying the condition
for all
then one of the following holds: (a)
(b)
(c) R/P is a commutative integral domain.
Moreover, some related results are also discussed. As consequences of our main theorems, many known results can be either generalized or deduced.
Acknowledgments
The authors are deeply indebted to the learned referees for their careful reading of the manuscript and constructive comments.