While reading about semisimple modules, the property every quotient is a subobject hence every submodule is projective gave me an idea to introduce projective modules directly as projections of free modules.
Call such modules which are projections of Free modules Projective!
Let me use this to develop the notion of projective modules. According to me the way projective modules are usually introduced, in terms of right exactness is strange.
Let $N$ be called projective if $N = \pi(A^n)$ for some projection map $\pi$, i.e., $\pi^2 = \pi$ (idempotent elements are natural examples). Projective modules are those which are images of such projection maps. This definition of Projective is in the right spirit of the word.
Now by properties of projection maps we get that $N$ is a direct summand of $A^n$. This we know is a standard equivalent criteria of projective modules. But let me also derive the right exactness criteria..it is a smooth proof. I don't see why projective modules are not introduced like this!
Let $N \oplus N' \cong A^n$.
Let $N \xrightarrow[]{f} M$. Also let $M_1 \xrightarrow[]{g} M $ be a surjective map. We need to find a $\bar{f}$ so that $g . \bar{f} = f$. we can take direct sum of each with $N'$ and the natural extensions of $f$ and $g$. NOw we get a $f_1 : A^n \to M_1 \oplus N'$ which is the factorisation. One can check that its restriction to $N$ is what we need.
Note that projections of projective modules are again projective.
Now my question is: What about modules which are projections of other modules or say projections of a fixed module and its direct sum copies? Can we still show some right exactness properties of such projections maybe for a special class of modules (obviously we can't for all since then the module is projective).
Say $M$ is a module. Then what can we say about its projections? If we take $M$ itself and $N_1$ and $N_2$ to be two quotients such that $N_2$ is a further quotient of $N_1$ then the quotient map from $M \to N_2$ factors through the quotient map from $N_1 \to N_2$. But can we say this for a class of modules and maps apart from just this? This can lead to a generalization of projective modules. Thank you.