On page 337 Dummit and Foote define a left R-Module (see attached) and then on page 339 they define and describe a $\displaystyle \mathbb{Z} $ module (see attached)

The definition of a $\displaystyle \mathbb{Z} $ module is as foillows:

Let R = $\displaystyle \mathbb{Z} $ module

Let A be any abelian group (finite or infinite) and write the operation of Aas +.

Make A into a $\displaystyle \mathbb{Z} $ module as follows:

For any $\displaystyle n \in Z $ and $\displaystyle a \in A $ define na as follows:

If n > 0 then na = a + a + ... ... + a (n times)

If n = 0 then na = 0

If n < 0 then na = -a - a - ,,, ,,, -a (-n times)

where 0 is the identity of the additive group.

D&F then write the following:

This definition of the action of the integers on A makes A into a $\displaystyle \mathbb{Z} $ module, and the module axioms show that this is the only possible action of Z on A making it a (unital) $\displaystyle \mathbb{Z} $ module. Thus every abelian group is a $\displaystyle \mathbb{Z} $ module.

My problem/question is with the statement: "the module axioms show that this is the only possible action of Z on A making it a (unital) $\displaystyle \mathbb{Z} $ module."

While this seems intuitive how do we know that this is true? How would one go about showing this is true?

Can anyone help with this issue?

Peter