On Simpleness of Semirings and Complete Semirings

TL;DR

This paper characterizes various types of simple semirings and their endomorphism structures, establishing invariance properties, describing specific classes, and confirming conjectures in the theory of semirings and semimodules.

Contribution

It provides a comprehensive classification of simple (complete) semirings, explores their invariance under Morita equivalence, and confirms key conjectures in the field.

Findings

Characterization of simple endomorphism semirings of idempotent commutative monoids.

Morita invariance of simpleness properties in semirings.

Complete description of ideal-simple, artinian additively idempotent chain semirings.

Abstract

In this paper, among other results, there are described (complete) simple - simultaneously ideal- and congruence-simple - endomorphism semirings of (complete) idempotent commutative monoids; it is shown that the concepts of simpleness, congruence-simpleness and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; there are described congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering the relationship between the concepts of "Morita equivalence" and "simpleness" in the semiring setting, we have obtained the following results: The ideal-simpleness, congruence-simpleness and simpleness of semirings are Morita invariant properties; A complete description of simple semirings containing the infinite element; The representation theorem - "Double Centralizer Property" - for simple semirings; A complete description of simple semirings containing a projective minimal one-sided ideal; A characterization of ideal-simple semirings having either infinite elements or a projective minimal one-sided ideal; A confirmation of Conjecture of [Kat04a] and solving Problem 3.9 of [Kat04b] in the classes of simple semirings containing either infinite elements or projective minimal left (right) ideals, showing, respectively, that semirings of those classes are not perfect and the concepts of "mono-flatness" and "flatness" for semimodules over semirings of those classes are the same. Finally, we give a complete description of ideal-simple, artinian additively idempotent chain semirings, as well as of congruence-simple, lattice-ordered semirings.