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Module and submodule

  1. Apr 29, 2008 #1
    1. The problem statement, all variables and given/known data
    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.


    2. Relevant equations

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

    3. 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].
     
    Last edited: Apr 29, 2008
  2. jcsd
  3. Apr 29, 2008 #2
    I could not fix my post and so I posted it again. Can someone delete one post for me? Thanks!!!
     
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