Projective semimodules. (Q1771956)

By a semimodule is meant a right semimodule over a semiring \(R\). A semimodule \(F\) is free if it has a linearly independent set of generators. A semimodule \(P\) is projective if for any surjective homomorphism \(f\colon M\to N\) and any homomorphism \(g\colon P\to N\) of \(R\)-semimodules there exists a homomorphism \(h\colon P\to M\) such that \(fh=g\). It is known that a semimodule \(P\) is projective iff it is a retract of a free semimodule. In this paper a projective semimodule is represented as a retract of a direct sum of its countably generated projective retracts with zero intersection. A characterization by means of congruences is also included. The results are analogous to that by \textit{M. Ploščica} [Algebra Univers. 31, No. 1, 135-146 (1994; Zbl 0802.08007)] for projective algebras.