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


  • Ihsen Yengui Université de Sfax
  • Faten Ben Amor Université de Sfax


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


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).


M. Coste, H. Lombardi, and M.-F. Roy, Dynamical method in algebra: Effective Nullstellensätze. Annals of Pure and Applied Logic 111 (2001), 203-256.

L. Ducos, C. Quitté, H. Lombardi, and M. Salou, Théorie algorithmique des anneaux arithmétiques, de Prüfer et de Dedekind. J. Algebra 281 (2004), 604-650.

L. Ducos, S. Monceur, and I. Yengui, Computing the $V$-saturation of finitely generated submodules of $V[X]^m$ where $V$ is a valuation domain. J. Symb. Comp. 72 (2016), 196-205.

A. Hadj Kacem and I. Yengui, Dynamical Gröbner bases over Dedekind rings, J. Algebra 324 (2010), 12-24.

G. Havas and B.S Majewski, Extended gcd calculation. Proceedings of the Twenty-sixth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1995). Congr. Numer. 111 (1995), 104-114.

A.K. Lenstra, H.W. Lenstra Jr, and L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), 515-534.

I. Yengui, Dynamical Gröbner bases, J. Algebra 301 (2006), 447-458.




How to Cite

Yengui, I., & Ben Amor, F. (2021). Saturation of finitely-generated submodules of free modules over Prüfer domains. Armenian Journal of Mathematics, 13(1). Retrieved from