  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 are not necessarily equivalent for semimodules over an
arbitrary semiring. We study several of these notions, in particular the
e-projective semimodules introduced by the first author using his new notion of
exact sequences of 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.
