Example on Z-modules .... Dummit & Foote, Page 339 ....

[Math Amateur]

I am reading Dummit and Foote's book: "Abstract Algebra" (Third Edition) ...

I am currently studying Chapter 10: Introduction to Module Theory ... ...

I need some help with an aspect of Dummit and Foote's example on Z-modules in Section 10.1 Basic Definitions and Examples ... ... Dummit and Foote's example on Z-modules reads as follows:
https://www.physicsforums.com/attachments/8001In the above example we read the following:

" ... ... This definition of an action on the integers on \(\displaystyle A\) makes \(\displaystyle A\) into a \(\displaystyle \mathbb{Z}\)-module, and the module axioms show that this is the only possible action of \(\displaystyle \mathbb{Z}\) on \(\displaystyle A\) making it a (unital) \(\displaystyle \mathbb{Z}\)-module ... ... "Can someone please explain how/why the module axioms demonstrate that this is the only possible action of \(\displaystyle \mathbb{Z}\) on \(\displaystyle A\) making it a (unital) \(\displaystyle \mathbb{Z}\)-module ... ... ? How do we know it is the only possible such action ... ... ?Hope someone can help ...

Peter

 

[Opalg]

If $n>0$ then $n = 1+1+\ldots + 1$ (property of $\Bbb{Z}$). Therefore $$na = (1+1+\ldots + 1)a = 1a + 1a + \ldots + 1a = a+a+\ldots + a,$$ using the module axioms. So that is the only possible value for $na$.The cases $n=0$ and $n<0$ work in a similar way, starting from the facts that $0 = 1 + (-1)$ and $-n = (-1)n$.