Let $(R,\mathfrak m)$ be a Noetherian local ring.
Definition: $I$ is called locally complete intersection ideal if $I_p$ is a complete intersection for all $p\in V(I)$.
I want an example of an ideal $I$ satisfying the following three properties:
1) grade$(I)\geq 1$,
2) $I$ locally complete intersection ideal but not an $\mathfrak m$-primary and complete intersection ideal,
3) $I$ is not integrally closed.
Any suggestion or reference will be extremely helpful. Thank you in advance.