Correspondence between Abelian Groups and Z-Modules
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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 statements are equivalent:
- $G$ is the underlying group of $M$.
- $M$ is the $\Z$-module associated with $G$.
Homomorphisms
Let $G, H$ be abelian groups.
Let $f : G \to H$ be a mapping.
The following statements are equivalent:
- $f$ is a group homomorphism.
- $f$ 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:
- 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.