Commutativity results for semiprime rings with derivations

Abstract

ABSTRACT. We extend a result of Herstein co,cerning a derivation d on a prime ring R satisfying [d(x),d(y)] 0 for all x,y R, to the case of semiprime rings. An extension of this result is proved for a two-sided ideal but is shown to be not true for a one-sided ideal. Some of our recent results dealing with U*-and U**-derivations on a prime ring are extended to semiprime rings. Finally, we obtain a result on semiprime rings for which d(xy) d(yx) for all x,y in some ideal U.

On derivations and commutativity of prime rings with involution

“…This result was extended for semiprime rings in [8] by Daif. Further, for semiprime rings, Andima and Pajoohesh [3] showed that an inner derivation satisfying the above mentioned condition on a nonzero ideal of must be zero on that ideal.…”

On derivations involving prime ideals and commutativity in rings

“…In [10] Herstein proved that if a prime ring R of characteristic different from two admits a nonzero derivation d such that [d(x), d(y)] = 0 holds for all x, y ∈ R, then R is commutative. Daif in [8] extended this result for semiprime rings.…”

A Characterization of Derivations in Prime Rings with Involution

“…In [13], Herstein proved that if R is a prime ring of characteristic not two admitting a nonzero derivation d such that [d(x), d(y)] = 0 for all x, y ∈ R, then R is commutative. Further, Daif [10] showed that a 2-torsion free semiprime ring R admits a nonzero derivation d such that [d(x), d(y)] = 0 for all x, y ∈ I, where I is a nonzero ideal of R, then R contains a nonzero central ideal. In [15], Lanski prove that if L is a noncommutative Lie ideal of a 2-torsion free prime ring R and d, h are nonzero derivations of R such that [d(x), h(x)] ∈ C for all x ∈ L, then h = λd, where λ ∈ C. Very recently, the first author together with Dar [11] proved the following result: Let R be a prime ring with involution * of the second kind such that char(R) = 2.…”