B\'ezout domains and lattice-valued modules

TL;DR

This paper establishes a Feferman-Vaught type theorem for modules over a commutative Bézout domain, linking definable sets to localizations and the maximal spectrum, with implications for decidability in algebraic integer rings.

Contribution

It introduces a new theorem connecting module definability over Bézout domains with spectral and localization properties, enabling decidability results.

Findings

Feferman-Vaught type theorem for B-modules

Decidability results for algebraic integer rings

Analysis of definable sets via spectrum and localizations

Abstract

Let B be a commutative B\'ezout domain B and let MSpec(B) be the maximal spectrum of B. We obtain a Feferman-Vaught type theorem for the class of B-modules. We analyse the definable sets in terms, on one hand, of the definable sets in the classes of modules over the localizations of B by the maximal ideals of B, and on the other hand, of the constructible subsets of MSpec(B). When B has good factorization, it allows us to derive decidability results for the class B-modules, in particular when B is the ring of algebraic integers or its intersection with real numbers or p-adic numbers.