Definition:Z-Module Associated with Abelian Group/Definition 1

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

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

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 $x$.

Also see

 * Equivalence of Definitions of Z-Module Associated with Abelian Group