On the Commutativity of Prime Rings with One-Sided Ideals and Derivations

Abstract

This article investigates the commutativity of rings using certain properties of derivations on prime rings. Let R be a prime ring with center Z(R), and let I be a nonzero left (right) ideal of R. Suppose that d is a derivation on R satisfying the condition d(Z(R)) 6= 0. In this paper, we prove that if one of the following conditions holds: (i) d([x; y])[z; x] 2 Z(R) (ii) d([x; y])[y; x] 2 Z(R) (iii) d(xy)xz 2 Z(R) (iv) d(xy)zy 2 Z(R) (v) d(xy) [z; x] 2 Z(R) (vi) d(yx) z y 2 Z(R) for all x; y 2 I, then R is commutative. Moreover, we provide an example showing that R must be a prime ring.