ON JACOBSON RADICAL OF A SEMIRING

We introduce the notions of Jacobson radical of 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 semiring S and that of the right operator semiring R of S, which we use to obtain characterizations of Jacobson radical of 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 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, number of characterizations of semisimple semiring is obtained. For preliminaries of semirings, semirings, operator semirings of semi- ring and rings we refer to (4), (9), (1), (6) and references therein. Throughout this paper semiring is assumed to be with zero, the left unity, the right unity. It is also assumed that S semimodule is additively cancellative.