On f -derivations on polynomial modules

An additive mappings δ on R is called a derivation if δ(ab) = δ(a)b + aδ(b) for all a,b ∈ R. If δ is a derivation on a ring R, M and N are right R-modules and f is a right R-linear mapping from M to N, then an additive mapping d: M → N is called a (δ,f)-derivation if d(xa) = d(x)a + f(x)δ(a) for all x ∈ M and a ∈ R. If δ is determined, then the (δ,f)-derivation is written briefly as f-derivation. The aim of this paper is to initiate the study of f-derivations on polynomial modules. In fact, we induce the derivations of polynomial rings R[x] based on the derivations of rings. In addition, we define R[x]-linear mapping on a polynomial module over a polynomial ring based on R-linear mapping on modules. Finally, we induce the f-derivations on polynomial modules over a polynomial ring based on f-derivations of modules and apply this result to the semigroup rings.