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We prove that if a valuation domain V has Krull dimension ≤ 1 then for every finitely generated ideal I of V [ X ] the trailing terms ideal t t (I) of I (that is, the ideal generated by the trailing terms of the nonzero polynomials in I) is also finitely generated. Heinzer and Papick (1989) and then Lombardi, Schuster and Yengui (2012) have shown the analogous result for leading terms ideals. The proof we give is both simple and constructive. The same result is valid for Prüfer domains. [ABSTRACT FROM AUTHOR]

SoitVun domaine de valuation. Nous donnons un algorithme pour calculer une base duV-saturé d'un sous-module de type fini d'unV-module libre (avec une base éventuellement infinie). Nous l'appliquons pour calculer leV-saturé d'un sous-V[X]-module de type fini deV[X]n(n∈ ℕ*). Ceci permet enfin de calculer un système générateur fini pour les syzygies surV[X] d'une famille finie de vecteurs deV[X]k. We give an algorithm for computing theV-saturation of any finitely generated submodule ofV[X]n(n∈ ℕ*), whereVis a valuation domain. This allows us to compute a finite system of generators for the syzygy module of any finitely generated submodule ofV[X]k. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to find necessary and sufficient conditions for the transfer of the strong S, catenarian, residually Jaffard and totally Jaffard properties between the rings R(1) and R<1> for domains R of dimension ≤ 2. Moreover, we give a positive answer to a conjecture of S. Kabbaj in the case of 2-dimensional integrally closed strong S-domains; we prove that if R is a 2-dimensional integrally closed strong S-domain then R[1] is catenarian. [ABSTRACT FROM AUTHOR]