Some differential identities in prime $gamma$ rings

Abstract

Let $M$ be a prime $\Gamma$-ring and $U$ be a nonzero ideal of $M$.
An additive mapping $d:M\longrightarrow M,$ where $M$ is a $\Gamma$-ring, is called a derivation if for any $a,b\in M$ and$\alpha \in \Gamma$, $d(a\alpha b)=d(a)\alpha b+a\alpha d(b)$. In this paper, we investigate the commutativity of prime $\Gamma$-ring satisfying certain differential identities.