Let I := {a_1X + · · · + a_nXn : n ∈ N, a_i ∈ Z for all 1 ≤ i ≤ n}. Show that I is a prime ideal but not a maximal ideal of Z[X].

Question

Let $I := {a_1X + · · · + a_nX^n : n ∈ N, a_i ∈ \mathbb{Z} \,\, for \,\, all \,\, 1 ≤ i ≤ n}.$ Show that I is a prime ideal but not a maximal ideal of Z[X].

My attempt:

While showing that it is a prime ideal, I assume, A,B ∈ Z[X], on solving for AB ∈ I, we obtain that either $a_0$ or $b_0$ = 0, so either A or B ∈ I, where $ A=a_0+a_1X+\dots a_nX^n $ and $ B=b_0+b_1X+\dots b_nX^n $.

But while proving it is not a maximal ideal, I assume $ I \nsubseteq J \subseteq \mathbb{Z}[X] $, I am not able to reach anywhere. How should I proceed

Answer 1

Notice that all monomials $x^n$ with $n\ge 1$ becomes 0 in the ring $\mathbb{Z}[X]/I$. So we have

$\mathbb{Z}[X]/I \cong \mathbb{Z}$

which is a domain, but not a field.

Edit

If you just want to show that it is not maximal, consider your ideal where you allow even constant terms.