The title is probably misleading, but I wasn't quite sure how to boil my question down to one line.
My problem is this:
Assume I have two projective $R$-modules, $P$ and $Q$, and epimorphisms $\varphi: P \to M$ and $\psi : Q \to M$, where $M$ is any $R$-module. Since $P$ and $Q$ are projective, I know we have $f: P \to Q$ and $g: Q \to P$ such that $ \varphi = \varphi \circ (g \circ f) \text{ and } \psi = \psi \circ (f \circ g),$ but what does that tell me about the relationship between $P$ and $Q$?
I mean loosely speaking, I can see that with respect to some element $m$ of $M$, the "class" of elements that maps to $m$ from $P$ are isomorphic to the "class" of elements that maps to $m$ from $Q$ ($\varphi^{-1}(m) \cong \psi^{-1}(m)$ for all $m \in M$), but isn't this just the canonical isomorphism $P/\ker{\varphi} \cong M \cong Q / \ker{\psi}$ (or am I wrong here?), we get from extending $P \to M \to 0$ to the obvious short exact sequence $0 \to \ker{\varphi} \to P \to M \to 0$.
Can I use the fact that $P$ and $Q$ are projective to say anything more specific about the relation between $P$ and $Q$ (for example that they are isomorphic)? Or perhaps about the relation between the kernels $\ker{\varphi}$ and $\ker{\psi}$?
Thanks!