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.

The Gröbner ring conjecture in one variable

Schuster, Peter Michael;
2012-01-01

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.
Bezout domain, valuation domain, semihereditary ring, Gröbner ring conjecture, constructive mathematics
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/927868
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