Basic Question on Modules - Dummit and Foote Chapter 10

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The discussion centers on the concept of module theory as presented in Dummit and Foote's Chapter 10. It highlights the assertion that an abelian group M can possess multiple R-module structures, even when the ring R remains constant. This complexity arises from the nature of the ring action, specifically through the ring-homomorphism $R \to \text{End}_{\Bbb Z}(M)$. The injectivity of this homomorphism is not guaranteed, leading to various module interpretations.

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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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