On the ideal structure of certain semirings and compactification of topological spaces

Abstract

Introduction. Let X be a topological space, R(X) its family of real-valued continuous functions, and (M(X) its family of open subsets. When it comes to reflecting the topological properties of X, there are many similarities between the ring and the lattice on R(X), and a certain strongly ordered structure on (M(X). In this connection the works of Stone [17], Alexandrov [1], Kaplansky [10], Hewitt [8], Milgram [12], Shirota [15], and Henriksen [7] should be mentioned. One similarity is that each has various families of which admit intrinsically defined, compact topologies. A natural domain in which to study this situation is the semiring. It was recently called to our attention that one such study has already been made. Slowiskowski and Zawadowski studied the space of maximal ideals in positive semirings [16]. Our principal results concern the family of R-ideals in a class of semirings suggested by the R-lattices of Shirota [15]. These semirings include various rings of continuous functions and the biregular rings (with identity) of Arens and Kaplansky [2 ], in addition to R-lattices. The notion of R-ideal is a generalization of the notions of lattice ideal and 0-ideal of Milgram [12]. The present paper and [16] seem to overlap very little, except in some of the applications. The author wishes to express his deep appreciation to the referee for his many helpful suggestions. In ?0, terms and conventions to be used throughout the work are given. The definitions of two particularly important relations, which are definable for any semigroup, appear in this section. These are the canonical order 0, and (in the language of relations) its square, the strong canonical order. ?1 contains the definition of R-ideal and Silov subset for any set S with transitive order R. Included among examples of Silov subsets are the Silov semigroups of Civin and Yood [5]. Some elementary properties of 0-ideals are proved in ?2. For an arbitrary relation R on S, the notion of prime-like (R) ideal is defined. This includes the familiar notion of prime ideal. The