TITLE: Need for projective modules
QUESTION [7 upvotes]: I wanted to ask why we require projective modules. After studying all the essential ingredients my guess is - 
Firstly, we worked with vector spaces (say modules over field $F$) (which are free modules)  and in that we could extend any basis of a submodule to get a basis for whole $V$ and thus property P
P- "any submodule of $V$ is a direct summand of $V$" is satisfied.
But in general for $R$-modules we could not express any submodule to be a direct summand, and thus we coined  semisimple rings whose definition is those rings $R$ for which every $R$ module $M$ satisfies P
But what inspired people to go for projective modules, what do they generalize?  
Two equivalent definitions of a projective module P are-

P is isomorphic to a direct summand of a free $R$ module.
every exact sequence of the form $$0 \to M'\to M\to P\to 0$$ splits.

I was looking for the inspiration that led to the study of projective modules and how do they help in simplifying studies of modules?

REPLY [3 votes]: Part of the motivation comes from topology, and is known as the Serre-Swan Theorem.
What the theorem says is that, for any compact Hausdorff space $X$, there is a one-to-one correspondence between vector bundles over $X$ and projective modules over the ring $C(X)$ of real-valued continuous functions on $X$.  Specifically, the set of all continuous sections of a vector bundle is a projective $C(X)$-module, and every projective $C(X)$-module has this form.
The same statement holds on a smooth manifold if you replace $C(X)$ by $C^\infty(X)$ (the ring of smooth real-valued functions on $X$) and “vector bundle” by “smooth vector bundle”.  There is also an algebraic geometry version over affine varieties involving the structure sheaf.