Generalized semirings of quotients

Abstract

A semiring is a non-empty set on which two associative binary operations, called addition and multiplication, are defined such that the multiplication distributes over the addition both from the left and from the right. In this paper all semirings considered will be assumed to have commutative operations as well as a zero. The semi-subtractive property for semirings is defined. Three aspects of ideal theory are investigated for semirings that have the semi-subtractive property. The concept of an irreducible ideal is generalized to semirings. It is shown that in a certain class of semirings every k-ideal is a finite intersection of primary ideals. Several consequences of this fundamental fact are proved. The concept of a quotient semiring of a semiring by a sub-semigroup of the semiring is developed for the class of semirings with the semisubtractive property. Extension of ideals to the quotient semiring is defined, and also contraction of ideals from the quotient structure. Properties of extension and contraction of ideals are investigated. Of particular interest is the behavior of prime and primary ideals under extension. Also, theorems concerning semi-isomorphisms and homomorphisms of quotient semiring structures are proved. Finally, the theory of the quotient structure provides a means of characterizing primary ideals in this class of semirings. The concepts of maximal prime divisors, minimal prime divisors, and prime divisors are extended to semirings. Theorems relating these concepts are proved. A necessary and sufficient condition for an ideal to be primary is given.