# The syzygy theorem for B\'ezout rings

## TL;DR

This paper extends Hilbert's syzygy theorem to Bézout rings, providing constructive methods for modules over Z, Z/nZ, and more general coherent strict Bézout rings, with applications to finitely generated modules.

## Contribution

It offers constructive versions of Hilbert's syzygy theorem for Bézout rings, expanding the scope beyond classical cases to more general rings with divisibility tests.

## Key findings

- Constructive versions of Hilbert's syzygy theorem for Z and Z/nZ.
- Extension of results to arbitrary coherent strict Bézout rings.
- Application to finitely generated modules with finitely generated modules of leading terms.

## Abstract

We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict B\'ezout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.