Centralizing Properties of $\left( \alpha,1\right) $ Derivations in Semiprime Rings

Abstract

Let R be a semiprime ring with center Z, S be a non-empty subset of R, $\alpha$ be an endomorphism on R and $d$ be an $\left( \alpha,1\right)$ derivation of R.
A mapping $f$ from R into itself is called centralizing on S if $ \left[ f(x),x\right] \in Z $, for\ all\ $ x\in S $.
In the present paper, we study some centralizing properties of $ (\alpha,1) $ derivations in semiprime rings one of the following conditions holds:
$\left( i\right) d\left( \left[ x,y\right] \right) =\left[ x,y\right] _{\alpha,1} $, for all \ $ x,y\in R$.
$\left( ii\right) d\left( \left[ x,y\right] \right) =-\left[ x,y\right] _{\alpha,1} $, for all \ $ x,y\in R $.
$ \left( iii\right) d\left( x\right) d\left( y\right) \mp xy \in Z $, for all\ $ x,y \in R $.
$\left( iv\right) d\left( xoy\right) =\left( xoy\right) _{\alpha,1} $, for all \ $ x,y\in R $ .
$\left( v\right) d\left( xoy\right) =-\left( xoy\right) _{\alpha,1} $, for all \ $ x,y\in R $.
Also we prove that $d$ is centralizing on R if $d$ acts as a homomorphism on R and $d$ is centralizing on S if $d$ acts as an antihomomorphism on R.