Relationship Between a Homoderivation and a Semi-Derivation

Abstract

Let $\wp$ be a ring. It is shown that if an additive mapping $\vartheta$ is a zero-power valued on $\wp$, then $\alpha:\wp\rightarrow\wp$ such that $\alpha=\vartheta+1$ is a bijective mapping of $\wp.$ The main aim of this study is to prove that $\vartheta$ is a homoderivation of $\wp$ if and only if $\vartheta:\wp\rightarrow\wp$ such that $\vartheta=\alpha-1$ is a semi-derivation associated with $\alpha$, where $\alpha:\wp\rightarrow\wp$ is a homomorphism of $\wp.$ Moreover, if $\vartheta$ is a zero-power valued homoderivation on $\wp,$ then $\vartheta$ is a semi-derivation associated with $\alpha$, where $\alpha :\wp\rightarrow\wp$ is an automorphism of $\wp$ such that $\alpha=\vartheta+1$.