On $*$-commuting mappings and derivations in rings with involution

Authors: NADEEM AHMAD DAR, SHAKIR ALI

Abstract: Let $R$ be a ring with involution $*$. A mapping $f:R\rightarrow R$ is said to be $*$-commuting on $R$ if $[f(x),x^*]=0$ holds for all $x\in R$. The purpose of this paper is to describe the structure of a pair of additive mappings that are $*$-commuting on a semiprime ring with involution. Furthermore, we study the commutativity of prime rings with involution satisfying any one of the following conditions: (i) $[d(x),d(x^*)]=0,$ (ii) $d(x)\circ d(x^*)=0$, (iii) $d([x,x^*])\pm [x,x^*]=0$ (iv) $d(x\circ x^*)\pm (x\circ x^*)=0,$ (v) $d([x,x^*])\pm (x\circ x^*)=0$, (vi) $d(x\circ x^*)\pm [x,x^*]=0$, where $d$ is a nonzero derivation of $R$. Finally, an example is given to demonstrate that the condition of the second kind of involution is not superfluous.

Keywords: Prime ring, semiprime ring, involution, additive mapping, $*$-commuting mapping, skew $*$-commuting mapping, derivation

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