Whether purely-prime ideals must be purely-maximal
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Must every purely-prime ideal of a commutative ring be purely-maximal? These are primality and maximality conditions taken within the lattice of pure ideals.
Why it matters and what counts as progress
Why it matters: The conjecture asks whether two notions coincide beyond the special ring classes where the implication is known.
What counts as progress: A general implication or a commutative ring with an explicitly classified pure-ideal lattice containing a purely-prime ideal that is not purely-maximal.
How the problem developed
AI contribution
A counterexample to a question on the maximality of purely-primes
For an integral domain k, the commutative ring R=k[X₁,X₂,…]/(Xᵢ(1−Xⱼ):i<j) has exactly the pure ideals 0, I=(x₁,x₂,…), and R. Thus 0 is purely-prime but not purely-maximal.
Sources
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Publication dates and editorial changes.
ChatGPT-assisted ring refutes a purely-prime ideal conjecture