Lie and Jordan Ideals in Prime Rings with Derivations

Abstract

In this paper derivations on Lie and Jordan ideals of a prime ring R are studied. The following results are proved, (i) Let R be a prime ring of characteristic not 2, and let U be a Lie or Jordan ideal of R. If d is a derivation defined on U, and if a is an element of the subring T(U), generated by U, or a is an element of R, according as U is a Lie or Jordan ideal of R, such that adu-0, for all u e U, then either a = 0 or du-0. (ii) Let a\, d2 be derivations defined for all u G U, and also for u2 and u3 if U is a Lie ideal of R, such that the iterate dxd2 is also a derivation, satisfying the same conditions as dx, d2. Let dx (u) G U, whether U is a Lie or Jordan ideal of R. Then, at least, one of dx(u) and d2(u) is zero, for all u G U. Introduction. Lemma 1 of Posner [1] states that if d is a derivation of prime ring R and a an element of R, such that ad(r) = 0, for all r G R, then either a = 0 or d is zero. Theorem 1 of Posner [1], which is a direct consequence of Lemma 1, states that if R is a prime ring of characteristic not 2 and if dx, d2 are derivations of R such that the iterate dx d2 is also a derivation, then at least one of dx, d2 is zero. The object of this paper is to generalize these results to Lie and Jordan ideals of R. All rings considered in this paper are associative. For definitions, see [2]. We prove the following results: Lemma. Let R be a prime ring of characteristic not 2 and let U be a Lie or Jordan ideal of R. If d is a derivation defined on U, and if a is an element of the subring T(U), generated by U, or a is an element of R, according as U is a Lie or