generalized inverse in the theory of projective module

Question

A module P over a ring R is projective which is an important topic in the theory of commutative ring due to its structural property of being a direct summand of free module. But my question is why we are studying about projective module and free module? If we will study about projective module, where we can use it in the future and how can we relate it in the theory of generalized inverses and minus partial order? Please tell me about the importance of projective module.

Answer 1

I don't know much ( ie. anything ) about generalized inverses, but I do know a bit about projective modules, at least over commutative rings, so I'll say something about that.

To me, I think of the importance coming from the relationship with vector bundles. It's hard to describe the relationship briefly if you don't have much background in algebraic geometry, but with a little handwaving: finite dimensional vector bundles on an affine space correspond one to one with ( coherent ) projective modules over the ring of functions on that space.

For smooth manifolds, Swan's theorem says that the space of sections of a vector bundle over a smooth compact manifold is a projective module over the ring of smooth functions on the manifold. To plug that in to your definition, that says that every vector bundle on a smooth compact manifold is a direct summand of a trivial bundle. That's pretty cool.

There are also "homological" reasons to study projective modules. This is because $Hom(P,-)$ is an exact functor exactly when $P$ is projective. That opens up a big tool for studying modules by studying a projective resolution of the module. Actually, you might more commonly use a free resolution, but the projective resolutions are nice and tight, to be really hand wavy, and so you can know things about them from more basic facts about the ring.

Ah, one more thing: Projective modules are the same as "locally free" modules. So, free modules are easy to work with and understand, and the next best thing is a module which is free after localizing at any prime ideal. This also kind of explains the relationship with vector bundles.