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On $*$-Commuting Mappings And Derivations In Rings With Involution, Nadeem Ahmad Dar, Shakir Ali

Turkish Journal of Mathematics

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 …