Z-Module Associated with Abelian Group is Unitary Z-Module

Theorem
Let $$\left({G, \circ}\right)$$ be an abelian group.

Let $$\circ$$ be the mapping from $$\mathbb{Z} \times G$$ to $$G$$ defined as in Product and Index Laws for Monoids: $$\forall n \in \mathbb{Z}: \forall x \in G: n \circ x = +^n x$$.

Then $$\left({G, +: \circ}\right)_{\mathbb{Z}}$$ is a unitary $\mathbb{Z}$-module.

This is called the $$\mathbb{Z}$$-module associated with $$G$$.