Prime Gamma Rings with Centralizing and Commuting Generalized Derivations

TL;DR

This paper proves that a prime Gamma-ring with a specific derivation property becomes commutative, extending understanding of the structure of such algebraic systems under certain derivation conditions.

Contribution

It establishes that a prime Gamma-ring with a centralizing and commuting generalized derivation on a left ideal must be commutative, revealing new structural constraints.

Findings

Prime Gamma-ring with certain derivation properties is commutative.

Generalized derivation conditions lead to commutativity in prime Gamma-rings.

Structural characterization of Gamma-rings under derivation assumptions.

Abstract

Let $M$ be a prime $\Gamma$-ring satisfying a certain assumption and $D$ a nonzero derivation on $M$. Let $f:M\rightarrow M$ be a generalized derivation such that $f$ is centralizing and commuting on a left ideal $J$ of $M$. Then we prove that $M$ is commutative.