For rings with unit there are at least three ways to define a projective module:
- The universal property, i.e. $P$ is projective if for any epimorphism $M\to N$ and any morphism $P\to N$ there exists a morphism $M\to P$ such that the diagram commutes.
- A projective module is a direct summand of a free module.
- Any epimorphism onto $P$ splits.
My question is what relation there is between these three conditions when the rings does not have a unit. Of course 1. implies 3., but what about the other directions? I'm most interested in "3. implies 1.?".
Edit: @t.b. has answered the 3. implies 1. In searching the web I found the claim that not even the rng (as the most basic free module) itself is projective. (see e.g. Anh, Marki: Morita equivalence over rings without identity and Arando Pino, Rangaswamy, Siles Molina: Weakly regular and self-injective Leavitt path algebras over arbitrary graphs.) So 2. seems not to imply 1. However I didn't find a counterexample in those papers. So, what is a counterexample for that?