Pushouts and e-Projective Semimodules

Projective modules play an important role in the study of the category of modules over rings and in the characterization of various classes of rings. Several characterizations of projective objects which are equivalent for modules over rings arenotnecessarily equivalent for semimodules over an arbitrary semiring. We study several of these notions, in particular thee-projectivity of semimodulesintroduced by the first author using his new notion ofexact sequencesof semimodules. As pushouts of semimodules play an important role in some of our proofs, we investigate them and give a constructive proof of their existence in a way that proved be very helpful.