Left Derivations and Strong Commutativity Preserving Maps on Semiprime $\Gamma$-Rings

TL;DR

This paper investigates derivations and endomorphisms in semiprime $ ext{Gamma}$-rings, showing that certain maps must be central or have specific forms, thereby extending previous results in ring theory.

Contribution

It proves that left derivations map into the center, and characterizes strong commutativity preserving derivations and endomorphisms in semiprime $ ext{Gamma}$-rings, extending prior work.

Findings

Left derivations map into the center of semiprime $ ext{Gamma}$-rings.

Strong commutativity preserving derivations imply the ring is commutative.

Endomorphisms with strong commutativity preservation have a specific additive form.

Abstract

In this paper, firstly as a short note, we prove that a left derivation of a semiprime $\Gamma$-ring $M$ must map $M$ into its center, which improves a result by Paul and Halder and some results by Asci and Ceran. Also we prove that a semiprime $\Gamma$-ring with a strong commutativity preserving derivation on itself must be commutative and that a strong commutativity preserving endomorphism on a semiprime $\Gamma$-ring $M$ must have the form $\sigma(x)=x+\zeta(x)$ where $\zeta$ is a map from $M$ into its center, which extends some results by Bell and Daif to semiprime $\Gamma$-rings.