  In this paper, we introduce homological structure theory of semirings and
CP-semirings---semirings all of whose cyclic semimodules are projective. We
completely describe semisimple, Gelfand, subtractive, and anti-bounded,
CP-semirings. We give complete characterizations of congruence-simple
subtractive and congruence-simple anti-bounded CP-semirings, which solve two
earlier open problems for these classes of semirings. We also study in detail
the properties of semimodules over Boolean algebras whose endomorphism
semirings are CP-semirings; and, as a consequence of this result, we give a
complete description of ideal-simple CP-semirings.
