/sci/ - Science & Math

Reposting from previous thread

Is it correct to say that a projective module is a module P such that every morphism onto it from any module has a right inverse that is also a morphism?

I'm trying to understand what a projective module is and why it's useful or important. I know the "lifting" definition but I don't get what it's trying to say, and I'm trying to work with its characterizations instead. For example I've seen projective modules be referred to as locally free modules, but I also don't get what's that's supposed mean since having a basis locally doesn't really seem to make too much sense, and I haven't seen someone define that property instead of just mention it (any textbook I should read that has that definition, by the way?). I've also seen the short split sequence "definition" (which we learned as a theorem), that states that a module P is projective iff every short exact sequence [math] 0 \longrightarrow A \longrightarrow B \longrightarrow P \longrightarrow 0 [/math] is split. Now, among all the characterizations of short split sequences, I remember that a short sequence is split iff the morphism from B to P has an inverse morphism to the right, and since that's supposed to happen for any module B, that's where I get my "definition" in the first paragraph, where I try to drop the context of short exact sequences and just mention surjective morphisms in general. Is this idea good enough, then? I'm a bit confused cause it seems like a way simpler form to understand its properties but no one states it like that. Also anyone has a better way to understand the idea of projective modules?