What was already known
Tarizadeh and Aghajani conjectured that every purely-prime ideal in a commutative ring is purely-maximal. Known special classes satisfied the claim; a noncommutative example did not settle it.
AI involvement
The first paper version credits ChatGPT Pro with finding the counterexample; the revision says Brian Conrad discovered it with ChatGPT assistance. Tarizadeh wrote the ring construction and proof.
The original version attributes finding the counterexample to ChatGPT Pro, while the paper's mathematical exposition and problem framing are human.
Importance
The construction refutes the general commutative-ring conjecture.
Limitations
The ChatGPT consultation transcript does not identify which proof steps came from the model. The available sources do not document a completed independent proof check.
Technical verification
Theorem 2.5 contains a detailed proof through separating maps to k and k[t] and classification of all pure ideals. The reviewed sources do not document a completed independent human or machine check; proof publication alone is not a checking method.
Human understanding
Tarizadeh explicitly explains why the quotient ring has exactly three pure ideals, which supplies a human explanation of the central construction.
Correction and publication history
The construction refutes the general commutative-ring conjecture.
Read this published revision