Extend the closed-point characterization to uncountable rings

Determine whether, for an uncountable ring R, a closed point U of the Ziegler spectrum Zg(R) necessarily has the ascending and descending chain conditions on pp-definable subgroups, equivalently whether the forward implication from closedness of {U} to these two chain conditions remains valid.

Background

For countable rings, the paper invokes Ziegler’s rank theorem to identify three equivalent properties of a point U in the Ziegler spectrum: the singleton {U} is closed, the module U satisfies both the ascending and descending chain conditions on pp-definable subgroups, and U is endofinite. The authors explain that this equivalence is the Cantor–Bendixson-rank-zero case of a deeper result relating the Cantor–Bendixson rank of basic open subsets of the Ziegler spectrum to the m-dimension of intervals in the lattice of pp-formulae. They explicitly note that it is unknown whether the forward implication from closedness to the two chain conditions continues to hold when the ring is uncountable.