Largest ring over which a particular abelian group becomes a module

Question

Suppose you have an abelian group $M$. Then $\operatorname{End}(M)$ is a ring with identity under point-wise addition and composition of functions. We can consider $M$ as an $\operatorname {End}(M)$-module by defining $\ \ f.m=f(m)$. Now given any ring $R$ for which $M$ is a $R$-module I have a ring homomorphism $\phi:R \to \operatorname{End}(M)$. Does this imply that $\operatorname{End}(M)$ is the "biggest" ring over which M is a module?

Answer 1

Sort of, but consider this: if $M$ is an $R$ module, then it's also an $R\times R$ module, and an $R^3$ module, and an $R^n$ module, and an $R\times S$ module for any other ring $S$. You can really blow up the ring arbitrarily and still have a homomorphism into $End_\mathbb Z(M)$. Since that's the case, it doesn't seem very useful to think of it as being "the largest."

It might be better to say this:

$End_\mathbb Z(M)$ is the largest ring for which $M$ is a faithful module.

That way you get that every ring for which $M$ is a faithful module embeds into $End_\mathbb Z(M)$, and that is a more appropriate context to think of $End_\mathbb Z(M)$ as being something maximal.