Free and Finitely Generated Modules

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In summary, the conversation discusses the relationship between free and finitely generated modules. It is clarified that not all free modules are finitely generated and not all finitely generated modules are free. An example is given to demonstrate this. The conversation also poses a question about the freeness of a module quotient in the context of a principal ideal domain.
  • #1
Sudharaka
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Hi everyone, :)

Want to confirm my understanding about Free and Finitely Generated modules. I want to know whether the following ideas are correct. Thank you for all your help. :)

1) Is every free module a finitely generated module?

No. Because a free module may have an infinite basis. So we cannot say it's finitely generated. However if the free module has a finite basis it's finitely generated.

2) Is every finitely generated module a free module?

No again. If \(M\) is a \(R\)-module which is finitely generated by a set \(S=\{x_1,\,x_2,\,\cdots,\,x_n\}\subset M\) then for each element \(x\in M\) we have,

\[x=r_1 x_1+\cdots+r_n x_n\]

where \(r_1,\cdots,r_n\in R\).

However we don't know whether \(S\) is linearly independent. So generally \(M\) is not free.

Am I correct? :)
 
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  • #2
That looks correct.

An easy counter-example for (2) is given by the $\Bbb Z$-module $\Bbb Z_n$.

Clearly, we have $\{1\}$ as a generating set, but this is not a basis because:

$n.1 = 0$ but $n \neq 0$.

Something for you to think about:

Suppose $R$ is a principal ideal domain, and that $M$ is a finitely-generated $R$-module.

Let $T = \{m \in M: \exists r \in R^{\ast}\text{ with }r.m = 0\}$.

Is $M/T$ free? (This is a subtler question than might appear at first. Ask yourself: why do we insist $R$ be a PID?).
 

FAQ: Free and Finitely Generated Modules

1. What is a free module?

A free module is a type of module in abstract algebra that is generated by a set of elements, where each element can be multiplied by any scalar from a given ring. This means that a free module has a basis, which is a linearly independent set of generators that can be used to construct all elements in the module.

2. What is a finitely generated module?

A finitely generated module is a type of module that can be generated by a finite set of elements. This means that all elements in the module can be expressed as a linear combination of these generators. Finitely generated modules are important in algebraic structures such as vector spaces and modules over a ring.

3. What is the difference between a free module and a finitely generated module?

The main difference between a free module and a finitely generated module is that a free module can have an infinite basis, while a finitely generated module has a finite basis. This means that a free module has more elements and is more "flexible" in terms of generating different elements compared to a finitely generated module.

4. How are free and finitely generated modules used in mathematics?

Free and finitely generated modules are used in various areas of mathematics, including abstract algebra, linear algebra, and ring theory. They are also important in applications such as coding theory, cryptography, and algebraic geometry. Free modules, in particular, are used to study and classify different algebraic structures and provide a deeper understanding of their properties.

5. Can a module be both free and finitely generated?

Yes, a module can be both free and finitely generated. This type of module is called a free module of finite rank. It has a finite basis and can be generated by a finite set of elements. Free modules of finite rank are particularly useful in applications where the number of generators needs to be limited, such as in coding theory and cryptography.

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