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It seems like a common exercise to try and find rings which only satisfy some of the conditions in the definition of a Dedekind domain. Rings that satisfy exactly 2 of the three conditions were very interesting to try and invent, but to satisfy just this one condition seems even more interesting.

Is there an integral domain with Krull dimension $1$ which is neither Noetherian nor integrally closed?

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If $K\subset L$ is a field extension which is not finite and $K$ is not algebraically closed in $L$, then $K+XL[X]$ is such an example.

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