Notes on Commutativity of Prime Rings

Abstract

Let R be a prime ring with center Z(R), J a nonzero left ideal, $\alpha$ an automorphism of R and R admits a generalized $(\alpha ,\alpha )$-derivation F associated with a nonzero ${(}\alpha ,\alpha {)}$-derivation d such that $d(Z(R))\ne (0)$. In the present paper, we prove that if any one of the following holds: $\textit{(i)}$ $F([x,y])-\alpha ([x,y])\in Z(R)$ (ii) $F([x,y])+\alpha ([x,y])\in Z(R)$ (iii) $F(x \circ y)-\alpha (x \circ y)\in Z(R)$ (iv) $F(x \circ y)-\alpha (x \circ y)\in Z(R)$ for all $x,y\in J$, then R is commutative. Also some related results have been obtained.

The paper is supported by the Anhui Provincial Natural Science Foundation (1408085QA08) and the Key University Science Research Project of Anhui Province (KJ2014A183) and also the Training Program of Chuzhou University (2014PY06) of China.