# Decidability of the theory of modules over Pr\"ufer domains with dense value groups

## TL;DR

This paper establishes algebraic conditions under which the theory of modules over certain Pr"ufer domains, especially Bézout domains with dense value groups, is decidable, providing both sufficient and necessary conditions.

## Contribution

It identifies algebraic criteria that guarantee decidability of module theories over Pr"ufer domains with dense value groups, including necessary conditions for Bézout domains.

## Key findings

- Decidability conditions for modules over Pr"ufer domains.
- Necessary and sufficient conditions for Bézout domains.
- Application to domains with dense value groups.

## Abstract

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Pr\"ufer (in particular B\'ezout) domains whose localizations at maximal ideals have dense value groups. For B\'ezout domains, these conditions are also necessary.