Prime Ideal that isn't Principal

Question

I'm trying to see how maximal ideals, prime ideals, and principal ideals are related for an arbitrary domain $D$. I know that every maximal ideal is a prime ideal. I know that principal ideals are not a subset of prime ideals (and hence not of maximal ideals) by the example $\langle x^2-1\rangle\subset\mathbb{C}[X].$ Now I'm trying to think of an example of a prime ideal that isn't principal. Are there any good examples?

Answer 1

$K[X;Y], \enspace K$ a field, is not a P.I.D. and the ideal $(X,Y)$ is prime, not principal.

In $\mathbf Z[X]$, the ideal $(2,X)$ is also prime, not principal.

Answer 2

Hint: consider $\langle 2,x\rangle$ in $\Bbb Z[x]$.