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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:

In the above example we read the following:

" ... ... This definition of an action on the integers on ##A## makes ##A## into a ##\mathbb{Z}##-module, and the module axioms show that this is the only possible action of ##\mathbb{Z}## on ##A## making it a (unital) ##\mathbb{Z}##-module ... ... "Can someone please explain how/why the module axioms demonstrate that this is the

Peter

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:

In the above example we read the following:

" ... ... This definition of an action on the integers on ##A## makes ##A## into a ##\mathbb{Z}##-module, and the module axioms show that this is the only possible action of ##\mathbb{Z}## on ##A## making it a (unital) ##\mathbb{Z}##-module ... ... "Can someone please explain how/why the module axioms demonstrate that this is the

__action of ##\mathbb{Z}## on ##A## making it a (unital) ##\mathbb{Z}##-module ... ... ? How do we know it is the__**only possible**__such action ... ... ?Hope someone can help ...__**only possible**Peter