If $R$ is commutative and $M$ is an $R$-module, we have a ring homomorphism $\alpha: R \to \operatorname{End}_{\mathsf{Ab}}(M)$. But is the image of $\alpha$ exactly $\mathrm{End}_{R\text{-}\mathsf{Mod}}(M)$?
Since $\alpha_r(m)=r\cdot m$, we have $$\alpha_r(s\cdot m)=r\cdot(s\cdot m)=(rs)\cdot m=(sr)\cdot m=s\cdot (r\cdot m)=s\alpha_r(m).$$
So, $\alpha_r$ is in fact an $R$-module endomorphism?