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

Toward Homological Structure Theory of Semimodules: On Semirings All of Whose Cyclic Semimodules Are Projective