# Are there any commutative rings in which no nonzero prime ideal is finitely generated?

Are there any commutative rings in which no nonzero prime ideal is finitely generated?

I feel like the example (or proof of impossibility) ought to be obvious, but I'm not seeing it.

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Fields, but that's just vacuously true. I suppose you want to discount those? –  Casteels Sep 1 '14 at 5:27

The standard example is $R := k[x_1, x_2, \ldots]/(x_1^2, x_2^2, \ldots)$. Then $R$ has a single prime ideal $\mathfrak{m} = (x_1, x_2, \ldots)$ which is not finitely generated ($\mathfrak{m}$ is the only prime since it is maximal, but every prime ideal of $R$ must contain $x_1^2, x_2^2, \ldots$ and hence $\mathfrak{m}$).