On Radicals of Semirings and Related Problems

TL;DR

This paper develops an external Kurosh-Amitsur radical theory for semirings, providing fundamental results on Jacobson and Brown-McCoy radicals of hemirings, including characterizations, descriptions, and invariance properties.

Contribution

It introduces a unified approach to radical theory in hemirings, characterizes various semisimple hemirings, and extends classical ring results to semiring context.

Findings

Unified internal and external radical approaches

Characterization of J-semisimple hemirings

Descriptions of R_{BM}-semisimple hemirings

Abstract

The aim of this paper is to develop an `external' Kurosh-Amitsur radical theory of semirings and, using this approach, to obtain some fundamental results regarding two Jacobson type of radicals --- the Jacobson-Bourne, J-, radical and a very natural its variation, J_{s}-radical --- of hemirings, as well as the Brown-McCoy, R_{BM}-, radical of hemirings. Among the new central results of the paper, we single out the following ones: Theorems unifying two, internal and external, approches to the Kurosh-Amitzur radical theory of hemirings; A characterization of J-semisimple hemirings; A description of J-semisimple congruence-simple hemirings; A characterization of finite additively-idempotent J_{s}-semisimple hemirings; Complete discriptions of R_{BM}-semisimple commutative and lattice-ordered hemirings; Semiring versions of the well-known classical ring results---Nakayama's and Hopkins Lemmas and Jacobson-Chevalley Density Theorem; Establishing the fundamental relationship between the radicals J, J_{s}, and R_{BM} of hemirings R and matrix hemirings M_{n}(R); Establishing the matric-extensibleness of the radical classes of the Jacobson, Brown-McCoy, and J_{s}-, radicals of hemirings; Showing that the J-semisimplicity, J_{s}-semisimplicity, and R_{BM}-semisimplicity of semirings are Morita invariant properties.