Does Int \((\mathbb Z)\) have the stacked bases property?
The author attempts to answer the question from the title posed by \textit{J. Brewer} and \textit{L. Klinger} [New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 231, 65-75 (2003; Zbl 1080.13508)], in some cases. One of the results is as follows: Theorem 4.5. The two-dimensional Prüfer domain \(R=\mathrm{Int}(\mathbb Z)=\{ f\in\mathbb Q[X] | f(\mathbb Z)\subseteq\mathbb Z\}\) has the stacked bases property iff, for every matrix \(B\in\mathcal M_{m\times 2}(R)\) (\(m\) any natural number) with the unit content, there exists a matrix \(C\in\mathcal M_{2\times 2}(R)\) such that \(BC\) has unit content and all \(2\times 2\) minors of \(BC\) are zero. It is known that a Prüfer domain either with dimension 1 or with finite character has the stacked bases property. Thus, the author has given another way (via the so-called UCS-property) one could look into the original question, namely by way of the existence of certain \(2\times 2\) matrices.
scientific article; zbMATH DE number 6116563