INDUCED $(\alpha,\beta)$-DERIVATIONS ON SEMIGROUP RINGS

Abstract

Derivations and their generalizations play an important role in the study of algebraic structures. Among these, $(\alpha,\beta)$-derivations extend the classical notion of derivations by incorporating ring endomorphisms. In this paper, we study the construction of $(\alpha,\beta)$-derivations on semigroup rings. Let $R$ be a ring and $S$ a semigroup, and consider the semigroup ring $R[S]$. By using function composition, we construct induced mappings on $R[S]$ arising from endomorphisms and $(\alpha,\beta)$-derivations on $R$. We prove that every ring endomorphism of $R$ naturally induces an endomorphism on the semigroup ring $R[S]$. Moreover, if $\delta$ is an $(\alpha,\beta)$-derivation on $R$, then the induced mapping defines an $(\bar{\alpha},\bar{\beta})$-derivation on $R[S]$. These results provide a natural extension of generalized derivations from rings to semigroup rings and establish a framework for studying derivations on related algebraic extensions, including polynomial rings as a special case of semigroup rings.