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

TL;DR

This paper develops a homological framework for semirings, focusing on CP-semirings where all cyclic semimodules are projective, providing comprehensive classifications and solving open problems in the structure theory.

Contribution

It introduces homological structure theory for CP-semirings, offering complete classifications of various subclasses and solving open problems in the field.

Findings

Complete descriptions of semisimple, Gelfand, subtractive, and anti-bounded CP-semirings.

Characterizations of congruence-simple subtractive and anti-bounded CP-semirings.

Description of ideal-simple CP-semirings via Boolean algebra semimodules.

Abstract

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.