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

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 (Z as the main example).