On f-derivations on polynomial modules

Abstract

An additive mappings [Formula: see text] on [Formula: see text] is called a derivation if [Formula: see text] for all [Formula: see text]. If [Formula: see text] is a derivation on a ring [Formula: see text], [Formula: see text] and [Formula: see text] are right [Formula: see text]-modules and [Formula: see text] is a right [Formula: see text]-linear mapping from [Formula: see text] to [Formula: see text], then an additive mapping [Formula: see text] is called a [Formula: see text]-derivation if [Formula: see text] for all [Formula: see text] and [Formula: see text]. If [Formula: see text] is determined, then the [Formula: see text]-derivation is written briefly as [Formula: see text]-derivation. The aim of this paper is to initiate the study of [Formula: see text]-derivations on polynomial modules. In fact, we induce the derivations of polynomial rings [Formula: see text] based on the derivations of rings. In addition, we define [Formula: see text]-linear mapping on a polynomial module over a polynomial ring based on [Formula: see text]-linear mapping on modules. Finally, we induce the [Formula: see text]-derivations on polynomial modules over a polynomial ring based on [Formula: see text]-derivations of modules and apply this result to the semigroup rings.