An example of an ideal needed.

Question

Let $A\subseteq \Bbb{C}[X]$ be an ideal. I am looking for an example of $A$ such that $A\subsetneq I(Z(A))$.

Here $Z(A)$ is the set of zeroes that are satisfied by all polynomials in $A$ and $I(Z(A))$ is the ideal such that all polynomials contained in it satisfy $Z(A)$.

Answer 1

Consider the ideal of all polynomials that have a zero of multiplicity greater than 1 at some specific place, for example at 0.

Observe that it is then quite intuitive that the information about the multiplicty is lost by doing I(Z(A)), so that you get a different ideal then.

Answer 2

Take any prime ideal $P \subset \mathbb{C}[X]$ and $A := P^2.$