On V-Semirings and Semirings all of whose Cyclic Semimodules are Injective

TL;DR

This paper introduces V- and CI-semirings, characterizes their properties, and solves open problems by providing complete descriptions of certain classes of these semirings.

Contribution

It defines and studies V- and CI-semirings, characterizes their structure, and resolves two open problems regarding congruence-simple classes.

Findings

All Jacobson-semisimple V-semirings are V-rings.

Complete descriptions of bounded distributive lattices, Gelfand, subtractive, semisimple, and anti-bounded CI-semirings.

Solved two open problems on congruence-simple subtractive and anti-bounded CI-semirings.

Abstract

In this paper, we introduce and study V- and CI-semirings---semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and anti-bounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple anti-bounded CI-semirings which solve two earlier open problems for these classes of CI-semirings.