Tarizadeh’s Conjecture 5.8 on the maximality of purely-prime ideals

The conjecture asks whether every purely-prime ideal in a commutative ring is purely-maximal. The original paper presents this as an open problem and notes that a commutative counterexample seemed difficult to find.

Known results

• The implication holds for Gelfand rings.
• It holds for reduced $mp$-rings.
• A known noncommutative example does not resolve the commutative-ring conjecture.

August 2026 counterexample

Abolfazl Tarizadeh’s preprint claims a commutative ring $R=k[X_1,X_2,X_3,\ldots]/J$ with a purely-prime ideal that is not purely-maximal, disproving the implication. It attributes the result to ChatGPT Pro, but the counterexample remains unconfirmed because no independent verification or mathematical response was found.

Current status (as of August 2026): A commutative counterexample is claimed, but until the preprint is independently checked, the conjecture remains unresolved.