# Definition:Associated Z-Module Functor

## Definition

Let $\mathbf{Ab}$ be the category of abelian groups.

Let $\Z$ be the ring of integers and $\mathbf{Z-Mod}$ the category of unitary modules over $\Z$.

The associated Z-module functor $\mathbf{Ab} \to \mathbf{Z-Mod}$ is the covariant functor with

 Object functor: sends an abelian group to its associated Z-module Morphism functor: sends a mapping to itself