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I'm looking for some easy to prove examples of Noetherian domains which are not UFD but are integrally closed in their fraction field.

The common examples I know (like $\mathbb Q[x,y]/(x^2+y^2-1)$ or ring of integers in some algebraic number field) , all of which are Dedekind domains, for them either showing integral closedness, or showing non-UFD , is difficult. So I was wondering if there are any easier examples or easier proofs.

This is obviously an open ended question so there need not be (most definitely will not be) any unique answer.

Thanks in advance.

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    $\begingroup$ The obvious example that comes to mind is $\mathbb{C}[x,y,z,w]/xy-zw$. This is 3-dimensional. Standard 2-dimensional examples are $\mathbb{C}[x,y,z]/z^n-xy$, for $n\geq 2$. All are integrally closed, but not UFD. $\endgroup$ – Mohan Jan 8 '20 at 9:12
  • $\begingroup$ @Mohan: how would a proof these examples go ? $\endgroup$ – user102248 Jan 8 '20 at 9:31
  • $\begingroup$ Here's the DaRT query which happened to already have both. $\endgroup$ – rschwieb Jan 8 '20 at 21:25
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This is an elaboration of the examples in Mohan's comment (to answer your comment).

In general, integral closedness and UFDness are hard to prove or disprove. But once you put some conditions, they are a couple of standard tools. All three examples mentioned above can be done by this approach.

Assumption: $R$ is a Noetherian domain, which is a quotient of an affine space (over a perfect field) by a single equation.

Serre's criterion for integral closeness: $R$ is integrally closed if $R$ satisfies $(R_1)$ and $(S_2)$, cf. https://en.wikipedia.org/wiki/Serre%27s_criterion_for_normality.

The condition $(S_2)$ is automatically satisfied. In order to check $(R_1)$, we use the Jacobian criterion. For instance, for $R = \mathbb{Q}[x,y]/(x^2+y^2-1)$, since $(2x,2y)R = R$, $R$ satisfies $(R_1)$.

For the UFD property, we have the following lemma.

A (Noetherian) integrally closed domain $R$ is a UFD if and only if every height 1 prime ideal of $R$ is principal.

For the examples Mohan mentioned, show that the ideal generated by x,z is height 1 but not principal (in both rings). The example of the circle, take the ideal generated by $x, y-1$.

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  • $\begingroup$ For those of us not in the know... what are $R_1$ and $S_2$??? $\endgroup$ – rschwieb Jan 8 '20 at 21:28
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    $\begingroup$ Hi @rschwieb, here is the link to the Wikipedia page, en.wikipedia.org/wiki/Serre%27s_criterion_for_normality. You can find more details about the conditions in commutative algebra books such as the books of Matsumura and Eisenbud. $\endgroup$ – Youngsu Jan 8 '20 at 21:33
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    $\begingroup$ @rschwieb Updated as suggested. Thanks. $\endgroup$ – Youngsu Jan 9 '20 at 21:19

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