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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?

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First we have endomorphisms of $M$ as an abelian group. These are the same as endomorphisms of $M$ as a $\mathbb{Z}$-module (every abelian group is canonically a $\mathbb{Z}$-module and preserving addition and subtraction is equivalent to preserving the $\mathbb{Z}$-module structure), so already $\text{End}(M) \cong \text{End}_{\mathbb{Z}}(M)$.

Some of these endomorphisms have the additional property of preserving the $R$-module structure. This picks out a subring of $\text{End}(M)$ called $\text{End}_R(M)$. The latter is a subring of the former because, by definition, an $R$-module endomorphism must first of all be an abelian group endomorphism.

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    Ok, I see this now; I just hadn't thought hard enough about what it means for something to be a morphism of $\mathbb{Z}$ modules vs $R$-modules.2012-05-31