The Gröbner ring conjecture in one variable (Q415466)

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.