Abstract
We prove that a valuation domain V has Krull dimension ≤ 1 if and only if for every finitely generated ideal I of V[X] the ideal generated by the leading terms of elements of I is also finitely generated. This proves the Gröbner ring conjecture in one variable. The proof we give is both simple and constructive. The same result is valid for semihereditary rings.
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Coquand T., Lombardi H.: Hidden constructions in abstract algebra (3) Krull dimension of distributive lattices and commutative rings. In: Fontana, M., Kabbaj, S.-E., Wiegand, S. (eds) Commutative ring theory and applications, Lecture notes in pure and applied mathematics vol 231, pp. 477–499. M. Dekker, New York (2002)
Coquand T., Lombardi H., Roy M.-F.: An elementary charaterization of Krull dimension. In: Corsilla, L., Schuster, P. (eds) From sets and types to analysis and topology: towards practicable foundations for constructive mathematics, Oxford University Press, UK (2005)
Ellouz A., Lombardi H., Yengui I.: A constructive comparison between the rings R(X) and R (X) and application to the Lequain-Simis induction theorem. J. Algebra 320, 521–533 (2008)
Hadj Kacem A., Yengui I.: Dynamical Gröbner bases over Dedekind rings. J. Algebra 324, 12–24 (2010)
Lombardi H.: Dimension de Krull, Nullstellensätze et Évaluation dynamique. Math. Zeitschrift 242, 23–46 (2002)
Lombardi H., Quitté C.: Constructions cachées en algèbre abstraite (2) Le principe local global. In: Fontana, M., Kabbaj, S.-E., Wiegand, S. (eds) Commutative ring theory and applications, Lecture notes in pure and applied mathematics vol 231, pp. 461–476. M. Dekker, New York (2002)
Lombardi H., Quitté C., Yengui I.: Hidden constructions in abstract algebra (6) The theorem of Maroscia, Brewer and Costa. J. Pure Appl. Algebra 212, 1575–1582 (2008)
Mines R., Richman F., Ruitenburg W.: A Course in Constructive Algebra, Universitext. Springer-Verlag, Berlin (1988)
Yengui I.: Dynamical Gröbner bases. J. Algebra. 301, 447–458 (2006)
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Lombardi, H., Schuster, P. & Yengui, I. The Gröbner ring conjecture in one variable. Math. Z. 270, 1181–1185 (2012). https://doi.org/10.1007/s00209-011-0847-1
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DOI: https://doi.org/10.1007/s00209-011-0847-1