Classes of Bézout domains that are not EDDs

Identify and characterize classes of Bézout domains that are not elementary divisor domains (EDDs).

Background

Beyond establishing positive criteria for the EDD property, a complementary unresolved direction is to describe and classify Bézout domains that fail to be EDDs.

The authors note that this question, posed explicitly in the literature, remains unanswered, underscoring the need for structural understanding of non-EDD Bézout domains.

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., 2024) in Introduction, concluding paragraph

I also do not know whether there are non-adequate, elementary divisor domains that are lcm splitting.

— LCM Splittings in GCD Domains  (2610.23799 - Preu, 16 Oct 2026) in Remark \ref{rmk:propertiesOfGCDDomains}, Section 2; Section 1? Final remarks and open questions, Section \ref{sec:final}

Recently, Hägg and Mörtberg announced an example of a B{e}zout domain that is not an elementary divisor domain -- I do not know whether it is lcm splitting.

— LCM Splittings in GCD Domains  (2610.23799 - Preu, 16 Oct 2026) in Remark \ref{rmk:propertiesOfGCDDomains}, Section 2; Section \ref{sec:final}