On $(δ,f)$-derivations and Jordan $(δ,f)$-derivations on modules

Abstract: Let $R$ be a ring with identity, $M,N$ right modules over $R$. An additive mapping $\delta$ from $R$ to $R$ is called derivation on ring $R$ if it satisfies the Leibniz condition. If $\delta$ is a derivation on $R$ and $f:M \rightarrow N$ is a module homomorphism over $R$, then an additive mapping $d:M \rightarrow N$ is called a $(\delta,f)$-derivation if it satisfies $d(xa)=d(x)a+f(x)\delta(a)$ for all $x \in M$ and $a \in R$. An additive mapping $\delta: R \rightarrow R$ is called Jordan derivation on ring $R$ if $\delta(x^2)=\delta(x)x+x\delta(x)$ for all $x \in R$, which is the generalization of derivation This paper presents generalization of Posner's First Theorem of $(\delta,f)$-derivation on $2$-torsion prime modules. It also provides a generalization of some results in case of $2$-torsion free prime modules from ring situation. Moreover, we introduce a Jordan $(\delta,f)$-derivation on modules and prove that every Jordan $(\delta,f)$-derivation on modules is a $(\delta,f)$-derivation on modules.