On the Spectrum of Nonstandard Dedekind Rings

TL;DR

This paper explores the structure of nonstandard Dedekind rings using nonstandard analysis, lattice theory, and valuation theory, revealing their ideal structure, spectrum, and relations to standard Dedekind rings.

Contribution

It introduces a detailed classification of ideals and spectra of nonstandard Dedekind rings, connecting nonstandard analysis with algebraic number theory.

Findings

Nonstandard Dedekind rings have a unique ideal structure with prime ideals contained in exactly one maximal ideal.

Valuation theory describes prime ideals, valuation rings, value groups, and residue fields in nonstandard Dedekind rings.

The spectrum and valuation spectrum of these rings are characterized using lattice and valuation group structures.

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.