Intersection of a prime ideal with a non-prime ideal

Question

Let $I_1$ be a prime ideal of $\mathbb{C}[X_1,...,X_n]$ and $I_2$ be another ideal of $\mathbb{C}[X_1,...,X_n]$ which is not a prime ideal. Assume further than $I_2 \not\subset I_1$. Under what additional conditions, can we conclude that $I_1 \cap I_2$ is not prime?

Answer 1

If we have $I_1 \subset I_2$, then the intersections equals $I_1$, which is prime.

Otherwise pick $x \in I_1 \setminus I_2$ and $y \in I_2 \setminus I_1$. We have $xy \in I_1 \cap I_2$ but none of the factors is contained in the intersection.

In fact the assumption $I_2 \not\subset I_1$ is not needed, since if $I_2 \subset I_1$ holds, the intersection is $I_2$, hence not prime. The important assumption is $I_1 \not\subset I_2$