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Math Help - socle in artinian module

  1. #1
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    socle in artinian module

    I had a presentation on a seminar recently. I said something like this

    If M is a right Artinian module and S is its socle, then

    S\subseteq_e M,

    because S=\bigcap\left\{N|N\subseteq_e M\right\} and the family of right submodules on the right-hand side has a smallest element since M is right Artinian, and it must be its intersection S. No one said I was wrong but I think I was. M being right Artinian only means that the family on the right has a minimal element.

    It's a true fact, proven differently in Lam. But what about what I said? Since no one protested during the seminar, I'm thinking maybe it's obvious how to finish the proof, but I don't know.
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  2. #2
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    Re: socle in artinian module

    Quote Originally Posted by ymar View Post
    I had a presentation on a seminar recently. I said something like this

    If M is a right Artinian module and S is its socle, then

    S\subseteq_e M,

    because S=\bigcap\left\{N|N\subseteq_e M\right\} and the family of right submodules on the right-hand side has a smallest element since M is right Artinian, and it must be its intersection S. No one said I was wrong but I think I was. M being right Artinian only means that the family on the right has a minimal element.

    It's a true fact, proven differently in Lam. But what about what I said? Since no one protested during the seminar, I'm thinking maybe it's obvious how to finish the proof, but I don't know.
    that's certainly not a valid proof! you've actually used what you're supposed to prove in your proof!! the intersection might not be a member of the family because an infinite intersection of essential submodules might not be essential. so the family might have a larger set as a minimal member.
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  3. #3
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    Re: socle in artinian module

    Thank you for the confirmation. I think the correct proof should be very simple, since Lam's was. But he uses a different definition of the socle and I want to use mine. I'll try to find the correct proof.
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