Essentially, I believe my question boils down to this: does the correspondence theorem at least partially hold for Lie algebras (or, indeed, any algebra)?

Lie algebras, . Let be some homomorphism of Lie algebras and let be an ideal of , . Then is it true that ?

This seems to be used in a proof in my lecture notes, but I can't work out why it holds (if, indeed, it does)!

Thanks in advance.