# Definition:Z-Module Associated with Abelian Group

## Definition

Let $\struct {G, *}$ be an abelian group with identity $e$.

Let $\struct {\Z, +, \times}$ be the ring of integers.

### Definition 1

The $\Z$-module associated with $G$ is the $\Z$-module $\struct {G, *, \circ}$ with ring action:

$\circ: \Z \times G \to G$:
$\tuple {n, x} \mapsto *^n x$

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 $\Z \to \map {\operatorname {End} } G$ equal to the initial homomorphism.