Let $A$ be an abelian group and $\mathrm{End}(A)$ its endomorphism ring. Then to give an abelian group $B$ the structure of a (left) $\mathrm{End}(A)$ module, we provide a morphism $\mathrm{End}(A)\to\mathrm{End}(B)$. Obviously $\mathrm{id}:\mathrm{End}(A)\to\mathrm{End}(A)$ gives $A$ the structure of an $\mathrm{End}(A)$ module.
Q: Does $A$ have in general any special properties as an $\mathrm{End}(A)$ module? Obviously it's not e.g. the free $\mathrm{End}(A)$ module on a single generator or something (since that's $\mathrm{End}(A)$).
A related question is whether non-isomorphic abelian groups can have isomorphic endomorphism rings. I suspect they can (but I should definitely think about this question more), which rules out the possibility that $A$ can have any truly universal properties in the category of $\mathrm{End}(A)$ modules, since universal properties are unique up to isomorphism.