Structure theory of semirings

Abstract

It is the purpose of this study not only to obtain structural information about classes of semirings which closely resemble rings but also to provide a setting for the theory of semirings in general. Accordingly, semirings and near-semirings are defined as special cases of monoids with operators. Kernels of homomorphisms are characterized and a natural factor is studied. A symmetry condition is introduced which generalizes normality for subgroups of a group. Homomorphisms which represent reversible submonoids and subsemirings as cancellative and the embedding of such representations into monoids of differences are studied. An analogue of a commutator subgroup is developed for monoids, and it is shown that for a large class of semirings such a commutator semiring is contained in the annihilator. Next, intrinsic preorders and partial orders of monoids are studied as tools for structural investigation. Among the topics examined are positive precones, directed preorders and prelattices, the characterization of intrinsic partial orders and of monoids admitting them, monotone homomorphisms and order epimorphisms, and the extension of intrinsic preorders to monoids of differences. The chapter concludes with a decomposition theorem for Archimedean preordered semi rings and a characterization of a small class of semirings whose additive semigroup is Gaussian. It is shown that for a large class of semirings, the behavior of certain epimorphisms is determined by the homomorphisms they induce on the maximum subgroup and the conic factor of the semiring by this subgroup. After considering several other approaches to the problem of determining when a semi-isomorphism is actually an isomorphism, the type of an ideal in a halfring is defined to be the family of all ideals in the ring of differences whose intersection with the semiring is the original ideal and it is shown that every homomorphic image of a halfring with given kernel is isomorphic to the natural factor exactly when the type of that kernel is a singleton. After a study of semirings free of zero-divisors and of multiplicatively cancellative semirings, attention is turned to chain conditions on ideals. It is shown that the semiring of polynomials over a halfring H is Noetherian if and only if the ring of differences of H is Noetherian, a result which includes the classical Hilbert Basis Theorem. An analogue of the Artin-Wedderburn Theorem is also obtained. Finally, division semirings and halffields are studied. The analogue for halfrings of the structural role played by division rings in ring theory is filled by objects called parafields, right and left simple halfrings without zero-divisors, The arithmetic of parafields is studied, and an attempt is made at classifying subparafields of the rationals.