Basic Question on Modules - Dummit and Foote Chapter 10

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In summary, the statement made by D&F emphasizes that an abelian group M can have multiple R-module structures even if the ring R remains constant. This is because the ring action involved in defining the module may not be injective. Therefore, an $R/I$-module can also be considered as an $R$-module.
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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.
 

FAQ: Basic Question on Modules - Dummit and Foote Chapter 10

What is a module?

A module is a mathematical structure that is similar to a vector space, but instead of being over a field, it is over a ring. It is a set of objects that can be added and multiplied by elements of the ring, satisfying certain axioms.

What is the difference between a module and a vector space?

The main difference is that a vector space is over a field, while a module is over a ring. This means that the scalars in a vector space can only be multiplied by elements from a field, while the scalars in a module can be multiplied by elements from a ring, which is a more general structure.

Can you give an example of a module?

One example of a module is the set of all matrices over a ring, where the ring is the set of real numbers. The addition and scalar multiplication of matrices satisfy the axioms of a module.

What is a submodule?

A submodule is a subset of a module that is also a module. It is closed under addition and scalar multiplication by elements of the ring.

How are modules used in mathematics?

Modules are used in many areas of mathematics, including abstract algebra, representation theory, and algebraic geometry. They provide a general framework for studying linear algebraic structures over rings, and are essential in understanding more complex mathematical objects.

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