The Gröbner ring conjecture in one variable. (English) Zbl 1242.13012
In the main result of this paper, the authors prove that a valuation domain \(\mathbf V\) has Krull dimension \(\leq 1\) if and only if for every finitely generated ideal \(I\) of \(\mathbf V[X]\) the ideal generated by the leading terms of elements of \(I\) is also finitely generated. This proves the Gröbner ring conjecture in one variable, and also gives an example of a class of non-Noetherian rings satisfying this property. As a consequence, they show that the result is valid for semihereditary rings. Finally, they ask two open problems.
Reviewer: Avanish Kumar Chaturvedi (Noida)
MSC:
| 13C10 | Projective and free modules and ideals in commutative rings |
| 19A13 | Stability for projective modules |
| 03F65 | Other constructive mathematics |
Keywords:
Bézout domain; valuation domain; semihereditary ring; Gröbner ring conjecture; constructive mathematicsReferences:
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