Definition:Z-Module Associated with Abelian Group

Definition
Let $\left({G, *}\right)$ be an abelian group with identity $e$.

Let $\left({\Z, +, \times}\right)$ be the ring of integers.

Definition 1
The $\Z$-module associated with $G$ is the $\Z$-module $(G, *, \circ)$ with ring action:
 * $\circ : \Z \times G \to G$:
 * $(n, x) \mapsto *^n x$

where $*^n x$ is the $n$th power of $x$.

Definition 2
The $\Z$-module associated with $G$ is the $\Z$-module on $G$ with ring representation $\Z \to \operatorname{End}(G)$ equal to the initial homomorphism.

Also see

 * Z-Module Associated with Abelian Group is Unitary Z-Module
 * Correspondence between Abelian Groups and Z-Modules