Paradox in ring theory -- what am I missing?

Question

I saw somewhere the following exercise:

Give example of prime ideal in a ring which is not maximal

the answer was this:

Let $R$ be our Ring and $I$ ideal such $$ R = {Z}[{X}] $$ $$ I = (x) $$ which means $I$ generated by $x$ which is all the polynomials in $R$ with zero free coefficient.

It was shown then, by First isomorphism theorem that : $$Z[X] / I \cong Z$$

Now, because $Z$ is not a field, but rather only Integral domain, then according to the theorem $I$ is indeed Prime ideal which is not maximal ideal.

But ....

As we know $Z[X]$ is Principal ideal domain thus according to other theorem:
every nonzero prime ideal is maximal

So if $I=(x)$ is ideal in $R$ it means I is maximal.

Why I came to this contradiction, what am I missing ?

Answer 1

Whereas $\mathbb{Z}$ is a principal ideal domain, $\mathbb{Z}[X]$ is not: for instance, the ideal $(2,X)$ is not generated by a single element.