Nagata and quotient criteria for additional singularity classes

Determine whether the nearly Gorenstein, canonical trace radical, and Cohen–Macaulay with canonical module properties satisfy the Nagata criterion or the quotient condition.

Background

Beyond bounded-codimension complete intersections, the paper introduces three additional classes of Noetherian local rings: nearly Gorenstein rings ($\mathsf{nGor}$), canonical trace radical rings ($\mathsf{CTR}$), and Cohen–Macaulay rings with a canonical module ($\mathsf{CMC}$). They form a hierarchy extending from Gorenstein rings toward Cohen–Macaulay rings.

The paper indicates that resolving the criteria for these classes requires understanding the behavior of their canonical modules. Whether any of these three properties satisfies either (NC) or (QC) is explicitly left unresolved.