$R$ is a commutative ring with $1$. Suppose every maximal ideal is finitely generated. Is this ring Noetherian? Equivalently, is every prime ideal finitely generated?
2026-03-27 16:21:44.1774628504
If every maximal ideal is finitely generated is the ring Noetherian?
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No.
A counterexample is the ring ${\cal O}(D)$ of the holomorphic functions defined on a domain $D\subset\Bbb C$. The maximal ideals are the ideals $\{(z-a){\cal O}(D)\}$ for $a\in D$ (which are principal), but there are ideals which are not finitely generated.
For instance, the ideal $I=\{\sin(nz)\}_{n\in\Bbb N}$ in ${\cal O}(\Bbb C)$ is proper (it is contained in $z{\cal O}(\Bbb C)$), but not finitely generated: look at the zero set of the elements in $I$.