Some Results for Endomorphisms in Prime Rings

Abdelkarim Boua

In this article, we present some commutativity theorems for a prime ring $\mathcal{R}$ equipped with endomorphisms $\alpha$, $\beta$, $\gamma$ and $\delta$ satisfying any one of the following identities: \begin{enumerate} em[(1)] $\:[\alpha(x), \beta(y)]+\gamma([x, y])+\delta(x\circ y)=0$ for all $x, y\in \mathcal{R};$ em[(2)] $\:\alpha(x)\circ \beta(y)+\gamma([x, y])=0$ for all $x, y\in \mathcal{R}$. \end{enumerate} Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.