# Centralizing Properties of $\left( \alpha,1\right)$ Derivations in Semiprime Rings Abstract views: 67 / PDF downloads: 48

## Authors

• G. Naga Malleswari Department of Mathematics, Sri Krishnadevaraya University, Anantapur, Andhra Pradesh, India
• S. Sreenivasulu Department of Mathematics, Government College(Autonomous), Anantapur, Andhra Pradesh, India
• G. Shobhalatha Department of Mathematics, Sri Krishnadevaraya University, Anantapur, Andhra Pradesh, India

## Keywords:

Semiprime rings, $\left( \alpha,1\right)$ derivation, centralizing mappings, homomorphism, antihomomorphism

## 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.

15-03-2020

## How to Cite

G. Naga Malleswari, S. Sreenivasulu, & G. Shobhalatha. (2020). Centralizing Properties of $\left( \alpha,1\right)$ Derivations in Semiprime Rings. International Journal of Mathematics And Its Applications, 8(1), 127–132. Retrieved from http://ijmaa.in/index.php/ijmaa/article/view/184

Research Article