286 research outputs found
On the Jacobson radical and semisimplicity of a semiring
Based on the minimal and simple representations, we introduce two
Jacobson-type Hoehnke radicals, m-radical and s-radical, of a semiring .
Every minimal (simple) -semimodule is a quotient of by a regular right
congruence (maximal) on such that is a maximal
-saturated right ideal in . 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
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