Tessellate S.T.E.M.S - Mathematics - College - Set 1 - Problem 4

Let $I$ be an ideal in the polynomial ring $\mathbb{Z}[X]$ , which is not maximal. Which of the following options are correct?

1. If the number of maximal ideals $M$ such that $I \subseteq M$ is finite, then $I$ is a principal ideal.
2. If the number of maximal ideals $M$ such that $I \subseteq M$ is infinite, then $I$ is not necessarily a principal ideal.
3. If the number of maximal ideals $M$ such that $I \subseteq M$ is $1$ , then $I \cap \mathbb{Z}$ is a principal ideal generated by a perfect square integer.

This problem is a part of Tessellate S.T.E.M.S.