# Some commutativity theorems for $*$-prime rings with $(\sigma,\tau)$-derivation

Document Type : Research Paper

Authors

1 Department of Mathematics,‎ ‎Aligarh Muslim University‎, ‎Aligarh‎, ‎202002, India.

2 Department of Mathematics,‎ ‎Aligarh Muslim University‎, ‎Aligarh‎, ‎202002, ‎India.

Abstract

‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(\sigma,\tau)$-derivation of $R$ with associated‎ ‎automorphisms $\sigma$ and $\tau$ of $R$‎, ‎such that $\sigma$‎, ‎$\tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{\sigma,\tau}=\{c\in‎ ‎R~|~c\sigma(x)=\tau(x)c~\mbox{for~all}~x\in R\}.$ In the present paper‎, ‎it is shown that if characteristic of $R$ is different from two and‎ ‎$[d(U),d(U)]_{\sigma,\tau}=\{0\},$ then $R$ is commutative‎. ‎Commutativity of $R$ has also been established in case if‎ ‎$[d(R),d(R)]_{\sigma,\tau}\subseteq C_{\sigma,\tau}.$

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### History

• Receive Date: 25 September 2014
• Revise Date: 31 July 2015
• Accept Date: 01 August 2015