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6 1 Let $A$ be a domain satisfying the following condition: If $mathfrak p_1,dots,mathfrak p_k$ are distinct nonzero prime ideals of $A$ , and if $m$ and $n$ are distinct elements of $mathbb N^k$ , then we have $$ mathfrak p_1^{m_1}cdotsmathfrak p_k^{m_k}nemathfrak p_1^{n_1}cdotsmathfrak p_k^{n_k}. $$ Is $A$ necessarily noetherian? This question is motivated by these outstanding answers of user26857 and Julian Rosen. user26857's answer shows that noetherian domains do satisfy the above condition, whereas Julian's answer shows that many non-noetherian domains do not satisfy it. commutative-algebra maximal-and-prime-ideals noetherian integral-domain share | cite | improve this question ...