I am looking for examples which shows that the inclusion $$\text{GCD}\subsetneq \text{Integrally Closed}$$ is strict.

I've found one pretty esoteric counterexample:

Let $R$ be the algebraic closure of $\Bbb C(x)$ and let $S$ be the integral closure of $\Bbb C[x]$ in $R$. Let $M$ be a maximal ideal of $S$, and let $\overline{\Bbb Q}$ be the algebraic closure of $\Bbb Q$. Then $\overline{\Bbb Q}+MS_M$ is integrally closed but not a GCD domain. (Example 2.10 here.)

Does anyone know of a more elementary example, or at least one which is more well-known?


Any Dedekind domain which is not a P.I.D. will be a counter-example.

Indeed, it can be shown that any noetherian gcd domain is a U.F.D., and a Dedekind domain which is a U.F.D. is actually principal.

Now Stark-Heegner's theorem gives the list of the $9$ imaginary quadratic fields with a principal ring of integers. $\mathbf Q(\sqrt{-5})$ is not in the list; so its ring of integers, which is $$\mathbf Z[\sqrt{-5}]\quad\text{since }-5\not\equiv 1\bmod 4,$$is a counter-example: it is integrally closed, but not a gcd domain.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.