Generalized derivations with central values on lie ideals LIE IDEALS

TL;DR

This paper investigates generalized derivations on lie ideals in prime and semiprime rings, establishing conditions under which these derivations are scalar multiplications or satisfy specific algebraic identities.

Contribution

It characterizes generalized derivations with central values on lie ideals, showing they are scalar multiplications or satisfy S4 under certain conditions.

Findings

H(x) = bx for some b in C under given conditions

Derivations satisfy S4 when certain centrality conditions hold

Existence of idempotent e such that H acts as scalar on eA

Abstract

Let R be a prime ring of H a generalized derivation and L a noncentral lie ideal of R. We show that if l^sH(l)l^t in Z(R) for all lin2 L, where s, t> 0 are fixed integers, then H(x) = bx for some b in C, the extended centroid of R, or R satisfies S4. Moreover, let R be a 2-torsion free semiprime ring, let A = O(R) be an orthogonal completion of R and B = B(C) the Boolean ring of C. Suppose ([x1; x2]sH([x1; x2])[x1; x2]t in Z(R) for all x1; x2 in R, where s, t> 0 are fixed integers. Then there exists idempotent e in B such that H(x) = bx on eA and the ring (1-e)A satisfies S4.