# Correspondence between Abelian Groups and Z-Modules

## Theorem

### Bijection

Let $\Z$ be the ring of integers.

Let $G$ be an abelian group.

Let $M$ be a unitary module over $\Z$.

The following are equivalent:

1. $G$ is the underlying group of $M$.
2. $M$ is the $\Z$-module associated with $G$.

### Homomorphisms

Let $G, H$ be abelian groups.

Let $f : G \to H$ be a mapping.

The following are equivalent:

1. $f$ is a group homomorphism.
2. $f$ is a $\Z$-module homomorphism between the $\Z$-modules associated with $G$ and $H$.

### Isomorphism of categories

Let $\Z$ be the ring of integers.

Let $\mathbf{Ab}$ be the category of abelian groups.

Let $\mathbf{\mathbb Z-Mod}$ be the category of unitary $\Z$-modules.

Then the:

In particular, $\mathbf{Ab}$ and $\mathbf{\mathbb Z-Mod}$ are isomorphic.