Centralizers on Prime and Semiprime Gamma Rings

TL;DR

This paper investigates the properties of left centralizers on semiprime and prime Gamma rings, establishing conditions under which these centralizers commute or are scalar multiples of each other.

Contribution

It provides new conditions under which centralizers on Gamma rings commute or are scalar multiples, extending the understanding of their structure in noncommutative settings.

Findings

Commutativity of centralizers under certain algebraic conditions.

Existence of scalar multiples relating different centralizers.

Results applicable to both semiprime and prime Gamma rings.

Abstract

Let $M$ be a noncommutative 2-torsion free semiprime $\Gamma$-ring satisfying a certain assumption and let $S$ and $T$ be left centralizers on $M$. We prove the following results: \\(i) If $[S(x),T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(x)]_{\alpha }$=$0$ holds for all $x\in M$ and $\alpha ,\beta \in \Gamma $, then $[S(x),T(x)]_{\alpha }$=$0$. \\(ii) If $S\neq 0 (T\neq 0)$, then there exists $\lambda \in C$,(the extended centroid of $M$) such that $T$=$\lambda \alpha S(S=\lambda \alpha T)$ for all $\alpha \in \Gamma $. \\(iii) Suppose that $[[S(x),T(x)]_{\alpha },S(x)]_{\beta }$=$0$ holds for all $x\in M$ and $\alpha ,\beta \in\Gamma $. Then $[S(x),T(x)]_{\alpha }$=$0$ for all $x\in M$ and $\alpha \in\Gamma $. \\(iv) If $M$ is a prime $\Gamma $-ring satisfying a certain assumption and $S\neq 0(T\neq 0)$, then there exists $\lambda \in C$, the extended centroid, such that $T$=$\lambda \alpha S(S=\lambda \alpha T)$.