Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2012; 27(1): 69-76

Printed March 1, 2012

https://doi.org/10.4134/CKMS.2012.27.1.69

Copyright © The Korean Mathematical Society.

On left $\alpha$-multipliers and commutativity of semiprime rings

Shakir Ali and Shuliang Huang

Aligarh Muslim University, Chuzhou Anhui

Abstract

Let $R$ be a ring, and $\alpha$ be an endomorphism of $R$. An additive mapping $H:R\longrightarrow R$ is called a left $\alpha$-multiplier (centralizer) if $H(xy)=H(x)\alpha(y)$ holds for all $x,y \in R$. In this paper, we shall investigate the commutativity of prime and semiprime rings admitting left $\alpha$-multipliers satisfying any one of the properties: (i) $H([x,y])-[x,y]=0$, (ii) $H([x,y])+[x,y]=0$, (iii) $H(x\circ y)-x\circ y=0$, (iv) $H(x\circ y)+x\circ y=0$, (v) $H(xy)=xy$, (vi) $H(xy)=yx$, (vii) $H(x^{2})=x^{2}$, (viii) $H(x^{2})=-x^{2}$ for all $x,y $ in some appropriate subset of $R$.

Keywords: ideal, (semi)prime ring, generalized derivation, left multiplier (centralizer), left $\alpha$-multiplier, Jordan left $\alpha$-multiplier

MSC numbers: 16N60, 16U80, 16W25