Prove: Module & Submodule Homework

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In summary, the conversation discusses the definitions of a pure submodule and a direct summand of a D_module. It also proposes a proof for showing that a direct summand is pure, and another proof for showing that if N is a pure submodule of a finitely generated torsion module M over a P.I.D D, then N is also a direct summand of M.
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Homework Statement


Suppose M is a D_module and N is a submodule. N is called pure iff for any y [tex]\in[/tex] N and a [tex]\in[/tex] D ax = y is solvable in N iff it is solvable in M. N is a direct summand of M iff there is a submodule K with [tex]M = N \oplus K[/tex]. Prove:
(1) If N is a direct summand, then N is pure.
(2) Suppose D is P.I.D and M is a finitely generated torsion module. IF N is pure, then N is a direct summand of M.

Homework Equations



I am not sure what it means for ax=y is solvable in M iff it is solvable in N

The Attempt at a Solution


(1) If M is a direct summand, then there is a submodule K with [tex]M = N \oplus K[/tex]. Let's suppose that ax=y is solvable in M for y [tex]\in[/tex] N and [tex]\in[/tex], then there is a [tex]\in[/tex] such that az=y. To prove that N is pure, one needs to prove that z [tex]\in[/tex] N. I do not know if this is what I am supposed to do and if so, I have no idea how to do it.
(2)Now D is a P.I.D and M is a finitely generated torsion module. Assume that N is pure. Let y [tex]\in[/tex] N and a [tex]\in[tex] D, then we have z [tex]\in[/tex] N such that az=y implies z [tex]\in[/tex] M. I do not know how to show that there is a submodule K of M such that [tex]M = N \oplus K[/tex].
 
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I could not fix my post and so I posted it again. Can someone delete one post for me? Thanks!
 

What is a module?

A module is a mathematical structure that generalizes the concept of vector spaces. It is a collection of elements that can be added, subtracted, and scaled by numbers, following specific rules.

What is a submodule?

A submodule is a subset of a module that is closed under the same operations as the larger module. In other words, it is a subset that is also a module itself.

How do you prove a module?

To prove a module, you must show that it satisfies the axioms of a module. These axioms include closure, associativity, commutativity, distributivity, and the existence of an identity element.

What is the purpose of proving a submodule?

The purpose of proving a submodule is to show that it is a valid subset of the larger module and follows the same rules and properties. This is important in understanding the structure of the module and its substructures.

What are some common techniques used to prove a module or submodule?

Some common techniques used to prove a module or submodule include direct proof, contradiction, and mathematical induction. The choice of technique often depends on the specific module or submodule being proven and the style of the proof.

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