On the Jacobson radical of a semiring

Abstract

Based on the minimal and simple representations of semirings, we introduce two Jacobson-type Hoehnke radicals, namely, m-radical and s-radical, of a semiring [Formula: see text]. Every minimal (simple) [Formula: see text]-semimodule is a quotient of [Formula: see text] by a regular right congruence (maximal) [Formula: see text] on [Formula: see text] such that [Formula: see text] is a maximal [Formula: see text]-saturated right ideal in [Formula: see text]. Thus the m(s)-radical becomes an intersection of a class of regular congruences. The m(s)-radical of the direct product of two semirings is the product of their individual [Formula: see text]-radicals.