# Decidability of the theory of modules over Pr\"ufer domains with infinite residue fields

## TL;DR

This paper establishes algebraic conditions under which the theory of modules over certain Pr"ufer domains, including Bézout domains with infinite residue fields, is decidable, extending previous understanding in algebraic logic.

## Contribution

It introduces algebraic criteria for decidability of module theories over Pr"ufer domains, generalizing prime radical relations, and proves necessity for Bézout domains.

## Key findings

- Decidability characterized by algebraic conditions
- Conditions are necessary and sufficient for Bézout domains
- Extends algebraic logic understanding to Pr"ufer domains

## Abstract

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Pr\"ufer (in particular B\'ezout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For B\'{e}zout domains these conditions are also necessary.