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.

Let $\circ$ be the mapping from $\Z \times G$ to $G$ defined as:


 * $\forall n \in \Z: \forall x \in G: n \circ x = *^n x$

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

The module $\left({G, *, \circ}\right)_\Z$ is called the $\Z$-module associated with $G$.

In $\Z$-Module Associated with Abelian Group is Unitary $\Z$-Module, it is shown that $\left({G, *, \circ}\right)_\Z$ is a unitary $\Z$-module.