A semiring extension

Abstract

A semiring is a non-empty set on which two associative binary operations, called addition and multiplication, are defined such that multiplication distributes over addition both from the right and from the left. In this paper all semirings considered will be assumed to have commutative addition and to contain a zero element. A halfring is a semiring which satisfies the additive law of cancellation. Ideals and Q-ideals of semirings are defined. Let I be an ideal of a semiring R. A quotient semiring R/I is defined. If I is a Q-ideal, a difference semiring R - I is defined and it is proven that R/I = R - I. The concept of an Everett extension ring is generalized to semirings. Let R and S be halfrings. A semiring E is an Everett extension semiring of R by S if R is an ideal of E and E/R congruent to S. A skew product S x R x R is defined. Certain subsets S + R + R (called Everett sums) of S x R x R are investigated. Necessary and sufficient conditions are established for these Everett sums to be semirings. Finally, it is proven that all the Everett extension semirings of a halfring R by a halfring S are (to within isomorphism) those Everett sums S + R + R which are halfrings and contain R as an ideal. An equivalence relation is defined on the set of all Everett extension semirings of R by S. Several theorems concerning equivalent Everett extension semirings are proved. The special case where E is an Everett extension semiring of R by S and R is a Q-ideal of E is studied. A skew product S + R is defined. It is proven that all the Everett extension semirings of a halfring R by a halfring S which contain R as a Q-ideal are (to within isomorphism) those skew products S + R which are halfrings. The theory of Everett extension rings is shown to follow directly from these results. It is also proven that every Everett extension semiring S + R + R may be represented as a subsemiring of an Everett extension ring S-bar + R-bar where S-bar and R-bar are certain extension rings of S and R respectively. The concept of double homothetisms and holomorphs of a ring is generalized to semirings. It is shown that the inner double homothetisms of all the Everett extension semirings of a halfring R induce all the double homothetisms of R. Finally, it is proven that a halfring R is a direct summand in all its Everett extension semirings if and only if it is a ring with unity.