Centrally Extended α -Homoderivations on Prime and Semiprime Rings

Abstract

We present a new type of mappings called centrally extended $\alpha$ -homoderivations of a ring $\Re$ (i.e., a map $H$ from $\Re$ into $\Re$ which satisfies $H\left(x+y\right)-H\left(x\right)-H\left(y\right)\in Z\left(\Re \right)$ and $H\left(xy\right)-H\left(x\right)H\left(y\right)-H\left(x\right)\alpha \left(y\right)-\alpha \left(x\right)H\left(y\right)\in Z\left(\Re \right)$ for any $x,y\in \Re$ ) where $\alpha$ is a mapping of $\Re$ and discuss the relationship between these mappings and other related mappings. Also, we study their effect on the center of a ring, and we give several outcomes on the commutativity of the ring.