MHB Basic Question on Modules - Dummit and Foote Chapter 10

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An abelian group M can possess multiple R-module structures even when both M and the ring R are fixed, due to the different ways the ring can act on the group. This variation arises from the potential for different ring homomorphisms from R to the endomorphism ring of M, which can be non-injective. The discussion highlights that the nature of the ring action is crucial in defining distinct module structures. An example illustrating this concept would clarify how different actions can lead to varied module interpretations. Understanding these distinctions is essential for grasping module theory as presented in Dummit and Foote.
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I am reading Dummit and Foote Chapter 10: Introduction to Module Theory.

After defining modules and giving some examples, D&F state the following:

"We emphasize that an abelian group M may have many different R-module structures even if the ring R does not vary ... ... "

I am puzzled by this statement ... surely if the abelian group M and the ring R is given, there is only one module being defined ...

Obviously I am wrong in this thought, but can someone please explain why I am wrong ...

Peter

EDIT * presumably the answer has something to do with the operation of the action involved ... but what exactly ...hopefully someone has an example that makes the whole thing clear ...
 
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Think of the ring action as a ring-homomorphism $R \to \text{End}_{\Bbb Z}(M)$.

There is no reason to suppose this homomorphism is injective.

Recall that any $R/I$-module has a natural interpretation as an $R$-module.
 
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