On page 337 Dummit and Foote define a left R-Module (see attached) and then on page 339 they define and describe a module (see attached)
The definition of a module is as foillows:
Let R = module
Let A be any abelian group (finite or infinite) and write the operation of Aas +.
Make A into a module as follows:
For any and 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 module, and the module axioms show that this is the only possible action of Z on A making it a (unital) module. Thus every abelian group is a 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) 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?