I had a presentation on a seminar recently. I said something like this
If is a right Artinian module and is its socle, then
because and the family of right submodules on the right-hand side has a smallest element since is right Artinian, and it must be its intersection No one said I was wrong but I think I was. 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.