Ashraf, M., Parveen, N. (2016). Some commutativity theorems for $*$-prime rings with $(\sigma,\tau)$-derivation. Bulletin of the Iranian Mathematical Society, 42(5), 1197-1206.

M. Ashraf; N. Parveen. "Some commutativity theorems for $*$-prime rings with $(\sigma,\tau)$-derivation". Bulletin of the Iranian Mathematical Society, 42, 5, 2016, 1197-1206.

Ashraf, M., Parveen, N. (2016). 'Some commutativity theorems for $*$-prime rings with $(\sigma,\tau)$-derivation', Bulletin of the Iranian Mathematical Society, 42(5), pp. 1197-1206.

Ashraf, M., Parveen, N. Some commutativity theorems for $*$-prime rings with $(\sigma,\tau)$-derivation. Bulletin of the Iranian Mathematical Society, 2016; 42(5): 1197-1206.

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

^{1}Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India.

^{2}Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India.

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

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 $[d(R),d(R)]_{\sigma,\tau}\subseteq C_{\sigma,\tau}.$