Generalized $\ast$-Lie ideal of $\ast$-prime ring

Authors: SELİN TÜRKMEN, NEŞET AYDIN

Abstract: Let $R$ be a $\ast$-prime ring with characteristic not $2,$ $\sigma, \tau:R\rightarrow R$ be two automorphisms, $U$ be a nonzero $\ast$-$\left( \sigma,\tau\right)$-Lie ideal of $R$ such that $\tau~$commutes with $\ast$, and $a,b$ be in $R.$ $\left( i\right)$ If $a\in S_{\ast}\left( R\right)$ and $\left[ U,a\right] =0$, then $a\in Z\left( R\right)$ or $U\subset Z\left( R\right) .$ $\left( ii\right)$ If $a\in S_{\ast}\left( R\right)$ and $\left[ U,a\right] _{\sigma,\tau}\subset$ $C_{\sigma,\tau}$, then $a\in Z\left( R\right) ~$or$~U\subset Z\left( R\right) .$ $\left( iii\right)$ If $U\not \subset Z\left( R\right)$ and $U\not \subset C_{\sigma,\tau}$, then there exists a nonzero $\ast$-ideal $M$ of $R$ such that $\left[ R,M\right] _{\sigma,\tau}\subset U$ but $\left[ R,M\right] _{\sigma,\tau}$ $\not \subset C_{\sigma,\tau}.$ $\left( iv\right)$ Let $U\not \subset Z\left( R\right)$ and~$U\not \subset C_{\sigma,\tau}.$ If $aUb=a^{\ast }Ub=0$, then $a=0$ or $b=0.$

Keywords: $\ast$-prime ring, $\ast$-$\left( \sigma,\tau\right)$-Lie ideal, $\left( \sigma,\tau\right)$-derivation, derivation

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