*-alpha-derivation on prime *-rings

Document Type : Research Paper


Department of Mathematics, Aligarh Muslim University, Aligarh, India


Let $\Re$ be an associative ring with involution $*$. An additive map $\lambda\rightarrow \lambda^{*}$ of $\Re$ into itself is called an involution if the following conditions are satisfied $(i) (\lambda\mu)^{*}=\mu^{*}\lambda^{*}$, $(ii) (\lambda^{*})^{*}=\lambda ~~ \mbox{for all}~ \lambda,\mu\in \Re$. A ring equipped with an involution is called an $*$-ring or ring with involution. The aim of the present paper is to establish some results on $*$-$\alpha$-derivations in $*$-rings and investigate the commutativity of prime $*$-rings admitting $*$-$\alpha$-derivations on $\Re$ satisfying certain identities also prove that if $\Re$ admits a reverse $*$-$\alpha$-derivation $\delta$ of $\Re$, then $\alpha\in Z(\Re)$ and some related results have also been discussed.