A Note on Differential Identities in Prime and Semiprime Rings

Abstract

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d (r ○ s)(r ○ s) + (r ○ s) d (r ○ s)ⁿ - d (r ○ s))^m for all r, s ϵ I, then R is commutative. (ii) If (d (r ○ s)(r ○ s) + (r ○ s) d (r ○ s)ⁿ - d (r ○ s)^m ϵ Z(R) for all r, s ϵ I, then R satisfies s₄, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.