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On the Jacobson radical of a semiring

Über Halbringe und Halbkörper. II

We introduce the notions of Jacobson radical of a semiring and semisimple semiring and characterize them via operator semirings. AMS Mathematics Subject Classification (2000): 16Y60, 16Y99, 20N10 characterizations. We obtain the relation between the Jacobson radical of a semiring S and that of the right operator semiring R of S, which we use to obtain characterizations of Jacobson radical of a semiring analogous to familiar results of the ring theory and semiring theory, (5). Then, with the help of the notion of subdirect sum of semiring, introduced at the outset in a similar way to that in ring, (6), and using the result that "a semiring S is semisimple if and only if its right operator semi- ring R is semisimple", a number of characterizations of semisimple semiring is obtained. For preliminaries of semirings, semirings, operator semirings of a semi- ring and rings we refer to (4), (9), (1), (6) and references therein. Throughout this paper a semiring is assumed to be with zero, the left unity, the right unity. It is also assumed that a S semimodule is additively cancellative.

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On jacobson radical of a semiring

This chapter discusses semirings and semifields. Among many concepts of universal algebras, semirings are those that originate from rings, by cancelling the assumption that ( R , +) has to be a group. Depending on how much other ring-like properties are also cancelled or added, various different concepts of semirings (S, +, ·) have been considered. Semirings with different properties have become important in theoretical computer science. A universal algebra S = (S, +, ·) with a nonempty set S and two binary operations, written as addition and multiplication, is called a “semiring” if (S, +) and (S,·) are arbitrary semigroups, such that a ( b + c ) = ab + ac and ( b + c ) a = ba + ca hold for all a, b, c ∈ S . In particular, ( S , +,·) is called a “proper semiring” if ( S , +) is not a group. Each distributive lattice is a semiring, clearly commutative and idempotent with respect to both operations. It has a zero or an identity if it is bounded from below or above, respectively.

Semirings and Semifields

Matrices over semirings

https://www.pnas.org/content/pnas/37/3/163.full.pdf

Samuel Bourne

Papers

The Jacobson Radical of a Semiring.