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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

__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 ...__

**only possible**Peter