Topological characterization of zero divisors in T"(X)

Characterize the zero divisors of the ring T"(X) in terms of the topology of X, such as via descriptions involving zero sets, cozero sets, or related topological structures on X.

Background

The paper provides a characterization of units in T"(X) (Theorem 2.11) and studies z-ideals, minimal ideals, and the socle of T"(X).

A direct topological characterization of zero divisors in T"(X) is posed as an open problem at the end.

References

We conclude this article by raising a few open problems for the reader. Can one characterise the zero divisors of the ring T"(X) using the topology on X?

— The ring of real-valued functions which are continuous on a dense cozero set (2502.15358 - Dey et al., 21 Feb 2025) in Introduction (end) and Section 6, item (1)