Example of an integral domain that is not integrally closed and having some localization which is also not integrally closed

Question

Can anyone show an example of integral domain that is not integrally closed and also has one of its localization with respect to a maximal ideal not integrally closed?

Answer 1

Let $k$ be a field, and consider the domain $A = k[x,y]/(x^2-y^3)\cong k[t^2,t^3]$ (the isomorphism is given by $x\mapsto t^3$ and $y\mapsto t^2$). Note that $t\notin A$. The field of fractions of $k[t^2,t^3]$ contains $t = t^3/t^2$, but $t$ is integral over $k[t^2,t^3]$, since it satisfies the monic polynomial $z^2 - t^2$. Hence $A$ is not integrally closed in its field of fractions.

For the localization part of your question, you can localize at the maximal ideal $(t^2,t^3)$ in $k[t^2,t^3]$ (corresponding to $(x,y)$ in $k[x,y]/(x^2-y^3)$). The same argument as above shows that the result is not integrally closed. Note that if you try to localize at any other maximal ideal, you'll end up inverting $t^2$ and getting $t$ in the localization, so the argument will break down.