Saturation of finitely-generated submodules of  free modules over Prüfer domains

Ihsen Yengui; Faten Ben Amor

doi:10.52737/18291163-2021.13.1-1-21

Authors

Ihsen Yengui Université de Sfax
Faten Ben Amor Université de Sfax

DOI:

https://doi.org/10.52737/18291163-2021.13.1-1-21

Keywords:

Valuation domains, Echelon form, Saturation, Abelian groups, Prüfer domains

Abstract

We propose to give an algorithm for computing the $R$-saturation of a finitely-generated submodule of a free module $E$ over a Prüfer domain $R$. To do this, we start with the local case, that is, the case where $R$ is a valuation domain. After that, we consider the global case ($R$ is a Prüfer domain) using the dynamical method. The proposed algorithm is based on an algorithm given by Ducos, Monceur and Yengui in the case $E=R[X]^m$ which is reformulated here in a more general setting in order to reach a wider audience. The last section is devoted to the case where $R$ is a Bézout domain. Particular attention is paid to the case where $R$ is a principal ideal domain ($\mathbb{Z}$ as the main example).

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