On the Jacobson radical of a semiring

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.