Additive maps on prime and semiprime rings with involution

Abstract

Let $R$ be an associative ring. An additive map $x\mapsto x^*$ of $R$ into itself is called an involution if (i) $(xy)^*=y^*x^*$ and (ii) $(x^*)^*=x$ hold for all $x\in R$. The main purpose of this paper is to study some additive mappings on prime and semiprime rings with involution. Moreover, some examples are given to demonstrate that the restrictions imposed on the hypothesis of the various results are not superfluous.