Axiomatizability of homological classes of semimodules over semirings

In this paper, we characterize semirings over which classes of projective, strongly projective, free, and injective semimodules are axiomatizable. Together with injectivity, we consider the concepts of Baer-injectivity and e-injectivity for semimodules over semirings and illustrate possible relationships between axiomatizabilities of the corresponding injective classes of semimodules, as well as characterize semirings over which the class of e-injective semimodules is axiomatizable.