Tarizadeh's Conjecture on the Maximality of Purely-Prime Ideals
Statement
Every purely-maximal ideal of a commutative ring is purely-prime, and the converse holds for several important classes of rings; Tarizadeh conjectured (Conjecture 5.8 of his earlier published paper) that in a commutative ring every purely-prime ideal is purely-maximal. False: there is a commutative ring with a purely-prime ideal that is not purely-maximal.
Context
A published conjecture in commutative algebra that stood for some years, disproved by its own poser with a model finding the counterexample. Narrow literature - between the week-old floor and the numbered-problem band.
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In the author's words: after years without progress on his own conjecture, "by using an advanced model of AI (ChatGPT Pro), a counterexample is found to this conjecture". The counterexample is the paper's content; the conjecture's own poser credits the model with finding it.
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