Math Help - socle in artinian module

1. 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.

2. Re: socle in artinian module

Originally Posted by ymar
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.

3. 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.