Rings and Algebras the Jacobson Radical of a Semiring

The concept of the Jacobson radical of a ring is generalized to semirings. A semiring is a system consisting of a set S together with two binary operations, called addition and multiplication, which forms a semigroup relative to addition, a semigroup relative to multiplication, and the right and left distributive laws hold. The additive semigroup of S is assumed to be commutative. The right ideal 7 of a semiring S is said to be right semiregular if for every pair of elements ii, i2 in 7 there exist elements ji and j2 in 7 such that ii + ji + i_ji + i2j2 = Ì2 + J2 + Ì1J2 + idi The Jacobson radical R of a semiring S is the sum of all the right semiregular ideals of S. I t is also the sum of all the left semiregular ideals of S. It is shown that the Jacobson radical of a semiring S has the two following important properties: (i) If R is the Jacobson radical of the semiring S, then the difference semiring S — R is semisimple, (ii) The Jacobson radical of a semiring S is a radical semiring. Thus the structure of an arbitrary semiring is reduced to the study of the structure of semisimple semirings and radical semirings. The paper concludes with a consideration of the Jacobson radical of a matrix semiring Sn . In the case S is a ring, the theory reduces to the Jacobson theory for arbitrary rings.