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?