## African Diaspora Journal of Mathematics

### On Commutativity of Prime Γ-Rings with $θ$-Derivations

#### Abstract

Let $M$ be a prime $\Gamma-$ring, $I$ a nonzero ideal, $\theta$ an automorphism and $d$ a $\theta-$derivation of $M$. In this article we have proved the following result: (1) If $d([x,y]_{\alpha})=\pm([x,y]_{\alpha})$ or $d((x\circ y)_{\alpha})=\pm((x\circ y)_{\alpha})$ for $x, y\in I; \alpha\in \Gamma$, then $M$ is commutative. (2) Under the hypothesis $d\theta=\theta d$ and $Char M\neq2$, if $(d(x)\circ d(y))_{\alpha}=0$ or $[d(x),d(y)]_{\alpha}=0$ for all $x, y\in I;\alpha\in \Gamma$, then $M$ is commutative. (3) If $d$ acts as a homomorphism or an anti-homomorphism on $I$, then $d=0$ or $M$ is commutative. Moreover, an example is given to demonstrate that the primeness imposed on the hypothesis of the various results is essential.

#### Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 20, Number 2 (2017), 39-44.

Dates
First available in Project Euclid: 17 May 2017

Huang, Shuliang; Rehman, Nadeem ur. On Commutativity of Prime Γ-Rings with $θ$-Derivations. Afr. Diaspora J. Math. (N.S.) 20 (2017), no. 2, 39--44. https://projecteuclid.org/euclid.adjm/1494986434