Find all prime ideals of $\mathbb{Z} / n\mathbb{Z} \ [x]$ 
Let $n=p_1^{r_1}p_2^{r_2}\ldots p_k^{r_k}$ be an integer with $p_i$ being its prime factors. Find all prime ideals of $\mathbb{Z} / n\mathbb{Z} \ [x]$ containing monic polynomials of degree one.   

I don't know how to approach this question.   
Any help will be appreciated.
Thanks.
 A: Basically, the problem is solved by using the Correspondence Theorem. For sake of notation, I will denote $\mathbb{Z}_n:=\mathbb{Z}/n\mathbb{Z}$ and $A:=\mathbb{Z}_n[x]$.
Let $q(x):=x-\bar{a} \in A$ be an arbitrary monic polynomial of degree one. By the Correspondence Theorem, there is a bijection between the prime ideals of $A$ containing $q(x)$ and the prime ideals of $A/\big(q(x)\big)$.
Now, if we consider the evaluation homomorphism $\phi: A \longrightarrow \mathbb{Z}_n$, given by
$$\phi\big(f(x)\big):=f(\bar{a}),$$
we get that $\text{Ker }\phi=\big(q(x)\big)$ and $\text{Im }\phi=\mathbb{Z}_n$. By the First Isomorphism Theorem, it follows that
$$A/\big(q(x)\big) \simeq \mathbb{Z}_n.$$
Therefore, the problem reduces to find the prime ideals of $\mathbb{Z}_n$. In turn, by the Correspondence Theorem, they are in bijection with the prime ideals of $\mathbb{Z}$ containing $n$. Since $\mathbb{Z}$ is a PID, its prime ideals are the ones generated by its prime elements (prime numbers, indeed). Hence, the prime ideals of $\mathbb{Z}$ containing $n$ are
$$(p_1),\dots,(p_k).$$
Going backwards in our correspondences, we get that the prime ideals of $A$ containing $q(x)$ are
$$\big(p_1,q(x)\big),\dots,\big(p_k,q(x)\big).$$
