Rings of continuous integer-valued functions and nonstandard arithmetic

Abstract

0. Introduction. In this paper rings of continuous integer-valued functions are studied, with particular attention paid to their maximal residue class domains. These domains correspond bijectively to minimal prime ideals, rendering the space of these ideals of particular interest. Since these domains are either the integers or are nonstandard models of the integers, questions about nonstandard arithmetic will also be considered. In ?1 the space of minimal prime ideals of C(X, Z), the ring of continuous functions from a nonempty Hausdorff space X into Z, the ring of integers, is showed to be homeomorphic to 6X (1.2), the Boolean space of the algebra of open-and-closed sets of X. The maximal ideal space of C(X, Z) is shown to map continuously onto 6X (1.3). The space, 6OX, of points of 3X that give rise to integer residue class domains, is studied in ?2. The map of X into boX strongly resembles the realcompactification injection [GJ]. A representation theorem of C(X, Z) over 3OX is also given (2.4). It is shown in ?3 that points in 3X 6OX give rise to Z, a nonstandard model of Z (3.1). Here some of the relevant background material in model theory is discussed. The algebraic theory of nonstandard arithmetic is studied in ?4. In ?5 we return to study Z, its maximal ideal space, and its quotient field Q, which is a nonstandard model of the rational field Q. In ?6, the most technical section of the paper, the valuations of Q associated with maximal ideals of Z are computed (6.3). The value groups that arise are analysed ((6.4), (6.5), and (6.6)), followed by some rather striking results in case the maximal ideal in question is principal. The ideals of Z are analyzed in ?7 along classical lines: i.e., we proceed from the study of maximal and prime ideals, through the study of primary ideals, to a decomposition theorem for ideals in terms of primary ideals (7.4). Ideals in C(X, Z) are decomposed in ?8, first into coprimary ideals (8.4), and then into primary ideals (8.9). In the process, the sets of maximal, prime, coprimary, and primary ideals of C(X, Z) are analyzed. In ?9 some model-theoretic results are obtained on the residue class fields of C(X, Z), the principal result being that any such field is elementarily equivalent