# Correspondence between Abelian Groups and Z-Modules/Isomorphism of Categories

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## Theorem

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:

- forgetful functor $\mathbf{\mathbb Z-Mod} \to \mathbf{Ab}$
- associated Z-module functor $\mathbf{Ab} \to \mathbf{\mathbb Z-Mod}$

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