Relationship Between a Homoderivation and a Semi-Derivation

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