On the Jacobson radical and semisimplicity of a semiring

TL;DR

This paper introduces Jacobson-type radicals for semirings based on minimal and simple representations, characterizes semisimple semirings as subdirect products of primitive semirings, and explores their structural properties.

Contribution

It defines new Jacobson-type radicals for semirings using regular congruences and characterizes semisimple semirings through primitive components and endomorphism semirings.

Findings

Semisimple semirings are subdirect products of primitive semirings.

Every s-primitive semiring can be embedded as a transitive subsemiring of endomorphisms.

The m- and s-radicals are intersections of regular congruences.

Abstract

Based on the minimal and simple representations, we introduce two Jacobson-type Hoehnke radicals, m-radical and s-radical, of a semiring $S$. Every minimal (simple) $S$-semimodule is a quotient of $S$ by a regular right congruence (maximal) $\mu$ on $S$ such that $[0]_\mu$ is a maximal $\mu$-saturated right ideal in $S$. Thus the m(s)-radical becomes an intersection of some regular congruences. Finally, every semisimple semiring is characterized as a subdirect product of primitive semirings; and every s-primitive semiring is represented as a 1-fold transitive subsemiring of the semiring of all endomorphisms on a semimodule over a division semiring.