# Module and submodule

## Homework Statement

Suppose M is a D_module and N is a submodule. N is called pure iff for any y $$\in$$ N and a $$\in$$ 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 $$M = N \oplus K$$. 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 $$M = N \oplus K$$. Let's suppose that ax=y is solvable in M for y $$\in$$ N and $$\in$$, then there is a $$\in$$ such that az=y. To prove that N is pure, one needs to prove that z $$\in$$ 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 $$\in$$ N and a $$\in[tex] D, then we have z [tex]\in$$ N such that az=y implies z $$\in$$ M. I do not know how to show that there is a submodule K of M such that $$M = N \oplus K$$.

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