Question: Does there exist an ideal in $\mathbb{Z}_4[x]$ which is prime but not maximal?
Thoughts: It seems to me that the ideal $(x)$ fails to be a prime ideal since $0 \in (x)= 2 * 2$ with $2 \notin (x)$ (alternatively because $\mathbb{Z}_4[x]/(x) = \mathbb{Z}_4$ which is not an integral domain). Likewise, the ideal $(2,x)$ within $\mathbb{Z}_4[x]$ is both prime and maximal because $\mathbb{Z}_4[x]/(2,x) = \mathbb{Z}_2$ is a field. Is my train of thought correct? At any rate, I'm not sure if ideal which is prime but not maximal exists, but in that case it would be nice if someone could prove that fact, or construct one to prove it exists.
Hint: Try to find an ideal such that the quotient is $\mathbb{Z_2}[x]$