On $^*$-differential identities in prime rings with involution

Abstract

Let $\mathcal{R}$ be a ring. An additive map $x\mapsto x^*$ of $\mathcal{R}$ into itself is called an involution if (i) $(xy)^*=y^*x^*$ and (ii) $(x^*)^*=x$ hold for all $x,y\in \mathcal{R}$. In this paper, we study the effect of involution $"*"$ on prime rings that satisfying certain differential identities. The identities considered in this manuscript are new and interesting. As the applications, many known theorems can be either generalized or deduced. In particular, a classical theorem due to Herstein [A note on derivation II, Canad. Math. Bull., 1979] is deduced.

Keywords

• prime ring,commutativity,involution,derivation,*-differential identities

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