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Semisimple modules

  1. Aug 15, 2008 #1
    Can somebody help me with the following proof:

    Let M be a semisimple module, say M = +_IS_i, where + denotes direct sum and S_i is a simple module.
    Then the number of summands is finite if and only of M is finitely generated.

    I have problem with understanding the proof of the following in my notes:

    if M is finitely generated then the number of summands is finite

    Can somebody help me in this argument.
     
  2. jcsd
  3. Aug 15, 2008 #2

    morphism

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    This follows immediately from the definition of a "finitely generated" module.
     
  4. Aug 15, 2008 #3
    I know that it is quite clear but still there is an argument which I dont understand and I somebody can help me with this, I will be greatful.
     
  5. Aug 15, 2008 #4

    morphism

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    What argument, exactly?
     
  6. Aug 15, 2008 #5

    mathwonk

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    try proving the contrapositive, that if M is any non zero module, that an infinite direct sum of copies of M cannot be finitely generated.

    recall the definition of direct sum, and in particular that only a finite number of summands can occur in each element of a direct sum.
     
  7. Aug 15, 2008 #6
    Ok the argument for this theorem in my notes which I dont understand is:

    Let M be finitely generated by u_1,...,u_r say. For each u_j we can find finitely many terms S_i whose sum contains u_j. Hence all the u_j are contained in the sum of a finite subfamily of the S_i and this family generates M so that I must be finite.

    I dont understand details of this so it will be good if you can help me with the details. Thanks.
     
  8. Aug 15, 2008 #7
    mathwonk, I have a proof of this which I dont understand.
     
  9. Aug 15, 2008 #8

    morphism

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    If you understand the appropriate definitions ("direct sum" and "finitely generated"), then the details will be crystal clear.
     
  10. Aug 15, 2008 #9
    Nut why is it true that this family generates M
     
  11. Aug 15, 2008 #10

    morphism

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    It generates it in the sense that its sum is M. (And this is true because M is generated, as a module, by u_1, ..., u_r.)
     
  12. Aug 15, 2008 #11
    Thanks alot
     
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