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Bézout domains of finite Krull dimension and the EDD property

Determine whether every Bézout domain of finite Krull dimension (particularly those with Krull dimension at least 2) is an elementary divisor domain (EDD).

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Background

The paper develops new criteria and refinements for when Hermite and Bézout rings are elementary divisor rings/domains (EDRs/EDDs), including unit-based and Pell-type conditions, but it does not settle broad classification questions.

One prominent question, cited from the literature (Fuchs–Salce [5]), asks whether the EDD property holds universally for Bézout domains of finite Krull dimension, especially for dimension at least 2. The authors explicitly state that this question remains unanswered.

References

The implicit and explicit questions raised in the literature, such as, “Is a B´ezout domain of finite Krull dimension [at least 2] an EDD?” (see [5], Ch. III, Probl. 5, p. 122), and, ‘What classes of B´ ezout domains which are not EDDs exist?’, remain unanswered.

— Matrix invertible extensions over commutative rings. Part III: Hermite rings (2405.01234 - Călugăreanu et al., 2 May 2024) in Introduction, concluding paragraph