abelian group as Z module

Question

How Would you prove that every abelian group can be understood as a Z-Module in a unique way?

I would guess that you would have to prove its bijective, but not sure how to go about this

Answer 1

Given an abelian group $G$ its set of endomorphisms $$ {\rm End}(G)=\{f:G\rightarrow G,\text{homomorphism}\} $$ is a ring under the usual operations of sum and composition, with unity the identity map.

Given any ring with unity $R$ there is a unique map of unital rings $$ \Bbb Z\longrightarrow R $$ given by $n\mapsto n\cdot1_R$ with the usual convention for $n<0$ (preserving the identity mandates $1_{\Bbb Z}\mapsto 1_R$). By applying this to ${\rm End}(G)$ you immediately see that $G$ has a unique structure of $\Bbb Z$-module.