Question regarding finitely generated modules

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In summary, the conversation discusses the equivalence of two statements for a finitely generated module P: 1) P is projective and 2) P is isomorphic to a direct summand of a free module. The conversation also mentions a diagram and the need to show 1 => 2. It is mentioned that P is isomorphic to the direct sum of a kernel and a subset K, and that K is isomorphic to M/ker(α). The speaker is stuck on showing P = K and realizes that their definition of projective was incorrect. The correct definition is given as: A module P is projective if and only if for any homomorphism f:M -> P that is onto, M is the direct
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I am supposed to show that the following are equivalent for a finitely generated module P:

1. P is Projective
2. P is isomorphic to direct summand of a free module
(There are 2 others but they refer to a diagram)

I am stuck on showing 1 => 2.

I know that since P is projective there is α: M -> P so that
M is isomorphic to ker (α) (direct sum) K,
where K is a subset of P.
Also since P is finitely generated P = Rx1 (direct sum) … (direct sum)Rxn.

I also know that K is isomophic to M/ker(α)

I believe I need to show that P = K, because then P would be isomorphic to a direct summand, but I don’t know how to show this.
 
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  • #2
I have recently noticed that my definition of Projective is incorect.
A module P is projective provided:
If f:M -> P is a homomorphism and onto then M = ker(f) (direct sum) K,
K contained in P.

Hopefully that makes my question easier.
 

Related to Question regarding finitely generated modules

1. What is a finitely generated module?

A finitely generated module is a mathematical structure that consists of a set of elements, along with operations on those elements, that satisfies certain properties. In particular, a finitely generated module is a module that can be generated by a finite number of elements.

2. What is the difference between a finitely generated module and an infinitely generated module?

The main difference between a finitely generated module and an infinitely generated module is the number of elements that are needed to generate them. A finitely generated module can be generated by a finite number of elements, while an infinitely generated module requires an infinite number of elements to generate it.

3. How are finitely generated modules used in mathematics?

Finitely generated modules are used in various areas of mathematics, including algebra, number theory, and topology. They are particularly useful in the study of vector spaces, where they are used to describe the structure of linear transformations and to prove theorems about vector spaces.

4. Can a finitely generated module be generated by a single element?

Yes, a finitely generated module can be generated by a single element. In fact, a module is finitely generated if and only if it can be generated by a finite number of elements, so a single element is sufficient to generate a finitely generated module.

5. What are some examples of finitely generated modules?

Some examples of finitely generated modules include finite abelian groups, polynomial rings, and vector spaces with a finite basis. In general, any module that can be generated by a finite number of elements is considered a finitely generated module.

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