Generalized derivations as a generalization of Jordan homomorphisms acting on Lie ideals and right ideals

TL;DR

This paper investigates generalized derivations on prime rings, focusing on their behavior on Lie and right ideals, and characterizes conditions under which certain algebraic identities hold within the ring's structure.

Contribution

It extends the understanding of generalized derivations acting on Lie and right ideals in prime rings, providing new conditions for algebraic identities involving these derivations.

Findings

Conditions under which H(u2)^n - H(u)^2n lies in the center for all u in a Lie ideal.

Characterization of when H(u2)^n - H(u)^2n equals zero for all u in a right ideal.

Insights into the structure of generalized derivations on prime rings.

Abstract

Let R be a prime ring with center Z(R) and extended centroid C, H a non-zero generalized derivation of R and n>1 a fixed integer. In this paper we study the situations: (1) H(u2)n-H(u)2n in C for all u in L, where L is a non-central Lie ideal of R; (2) H(u2)n - H(u)2n = 0 for all u in [I; I], where I is a nonzero right ideal of R.