On the Spectrum of Nonstandard Dedekind Rings

Abstract

The methods of nonstandard analysis are applied to algebra and number theory. We study nonstandard Dedekind rings, for example an ultraproduct of the ring of integers of a number field. Such rings possess a rich structure and have interesting relations to standard Dedekind rings and their completions. We use lattice theory to classify the ideals of nonstandard Dedekind rings. The nonzero prime ideals are contained in exactly one maximal ideal and can be described using valuation theory. The localisation at a prime ideal gives a valuation ring and we determine the value group and the residue field. The spectrum of a nonstandard Dedekind ring is described using lattices and value groups. Furthermore, we investigate the Riemann-Zariski space and the valuation spectrum of nonstandard Dedekind rings and their quotient fields.