Let $I$ be an ideal in a Noetherian quasi-unmixed local ring $(R,\mathfrak m).$

Let $q(I)=\overline{I}\cap I^{sat}$ where $I^{sat}=\cup_{n\geq 0}I:\mathfrak m^n.$ Then $q(I)$ is called relative integral closure of $I.$

Question Is there any example of a principal ideal $I$ (i.e. $I=(a)$) such that $I\neq q(I).$

I know that in normal domains, $I=q(I)$ for all principal ideals $I.$

Let $I$ be an ideal in a Noetherian quasi-unmixed local ring $(R,\mathfrak m)$ of dimension $d.$

Let $q(I)=\overline{I}\cap I^{sat}$ where $I^{sat}=\cup_{n\geq 0}I:\mathfrak m^n.$ Then $q(I)$ is called relative integral closure of $I.$

Question Is there any example of a principal ideal $I$ (i.e. $I=(a)$) such that $I\neq q(I)$ where $d\geq 2?$

I know that in normal domains, $I=q(I)$ for all principal ideals $I.$

Let $I$ be an ideal in a Noetherian quasi-unmixed local ring $(R,\mathfrak m).$

Let $q(I)=\overline{I}\cap I^{sat}$ where $I^{sat}=\cup_{n\geq 0}I:\mathfrak m^n.$ Then $q(I)$ is called relative integral closure of $I.$

Question Is there any example of a principal ideal $I$ (i.e. $I=(a)$) such that $I\neq q(I).$

I know that in normal domains, $I=q(I)$ for all principal ideal $I.$

Let $I$ be an ideal in a Noetherian quasi-unmixed local ring $(R,\mathfrak m).$

Let $q(I)=\overline{I}\cap I^{sat}$ where $I^{sat}=\cup_{n\geq 0}I:\mathfrak m^n.$ Then $q(I)$ is called relative integral closure of $I.$

Question Is there any example of a principal ideal $I$ (i.e. $I=(a)$) such that $I\neq q(I).$

I know that in normal domains, $I=q(I)$ for all principal ideals $I.$