$(\alpha, \beta)$-Derivation on the Polynomial Ring $R[x]$

Abstract

Derivations are important tools in the study of algebraic structures, providing a framework for analyzing ring and module behavior through differentiation-like operations. Among their generalizations, $(\alpha, \beta)$-derivations, defined via ring endomorphisms $\alpha$ and $\beta$—offer increased flexibility, particularly in non-commutative settings. While these derivations have been studied extensively on rings, their behavior on polynomial extensions remains unexplored. In this paper, we investigate $(\alpha, \beta)$-derivations on the polynomial ring $R[x]$, where $R$ is a ring equipped with a given $(\alpha, \beta)$-derivation. We propose a method to construct a derivation $(\alpha', \beta')$ on $R[x]$ from the original derivation on $R$, establish several of its fundamental properties, and analyze the relationship between the structures on $R$ and $R[x]$. Illustrative examples are provided to support the theoretical developments. This study offers a new perspective on the extension of generalized derivations to polynomial rings and contributes to the broader understanding of differential structures in algebra.