An example of an $m$-primary ideal in noetherian local domain

Question

Is there any example of a $m$-primary ideal $I$ in a noetherian local domain $(R, m)$ such that $I^2=mI\not=m^2 $?

Answer 1

Here's an example: Take $R = \mathbb{Q}[x,y,z,w]_{(x,y,z,w)}/(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$. One can check (e.g. in Macaulay2) that $R$ is a domain (with $\dim R = 1$). Let $m = (x,y,z,w)$ be the maximal ideal, and set $I = (x,y,z)$. Then the first $3$ relations in $R$ guarantee that $I^2 = mI$, while the last relation gives that $w^3 \in I^2$, so $m^3 \subseteq I$ and $I$ is $m$-primary, but in fact $w^2 \in m^2 \setminus I^2$.