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?
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.