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The Gröbner ring conjecture in one variable

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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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Correspondence to Henri Lombardi.

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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

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