Herstein’s theorem for generalized derivations in rings with involution

Year 2017, Volume: 46 Issue: 6, 1029 - 1034, 01.12.2017

Shakir Ali Abdul Nadim Khan Nadeem Ahmad Dar

Abstract

Let $R$ be an associative ring. An additive map $F:R\toR$ is called a generalized derivation if there exists a derivation $d$ of $R$ such that $F(xy)=F(x)y+xd(y)$ for all $x,y\in R$. In [7], Herstein proved the following result: If $R$ is a prime ring of $char(R)\neq 2$ admitting a nonzero derivation $d$ such that $[d(x),d(y)]=0$ for all $x,y\in R$, then $R$ is commutative. In the present paper, we shall study the above mentioned result for generalized derivations in rings with involution.

Keywords

Prime ring, involution, derivation, generalized derivation

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