Let $M$ be an $R$-module.
Considering $M$ as an abelian group, forgetting the $R$-action on $M$, we can unambiguously write $\mathsf{End}(M)$ to denote the endomorphism ring with addition defined as in $M$ and composition as the multiplication operation.
Next, considering $M$ as an $R$-module, we can write $\mathsf{End}_R(M)$ for the endomorphism ring of $R$-maps, that is, $\mathsf{End}_R(M) := \mathsf{Hom}_R(M,M)$.
Finally, we can consider the abelian group $M$ as a $\mathbb{Z}$-module and in this case we can write $\mathsf{End}_{\mathbb{Z}}(M)$ to denote the corresponding endomorphism rings.
So, we have three different endomorphism ring structures and I'm trying to understand precisely how they are related. Rotman, in Advanced Modern Algebra states that $\mathsf{End}_R(M)$ is in fact a subring of $\mathsf{End}_{\mathbb{Z}}(M)$ but I don't really see this. What's a good way to think about these three endomorphism rings and how they are related?