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.
Proof
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