On left $\alpha$-multipliers and commutativity of semiprime rings
Commun. Korean Math. Soc. 2012 Vol. 27, No. 1, 69-76
https://doi.org/10.4134/CKMS.2012.27.1.69
Printed March 1, 2012
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
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