Ideals of $\mathbb{Z}_n[x]$.

Question

What are the ideals of $\mathbb{Z}_n[x]$. Do all ideals of $\mathbb{Z}_n[x]$ are principal. I know if $R$ is an integral domain, then $R[x]$ is PID. ( Here, $\mathbb{Z}_n$ need not be ID ). Also, which the ideals of $\mathbb{Z}_n[x]$ are prime and maximal.

Answer 1

for each $p| n$, $(p)$ is a minimal prime ideal, thus you know all the prime and maximal ideals, they are of the form $$(p,f(x)), \qquad f(x)\in \Bbb{F}_p[x] \text{ irreducible}$$

If $I$ is an ideal and $n = ab,\gcd(a,b)=1$ then $$(I,a)(I,b)=(I^2,n,Ia,Ib)= (I^2,I(a,b))=(I^2,I)=I$$ thus $$I = \prod_{p^k\| n} (I,p^k)$$ A maximal ideal $(p,f(x))$ is principal iff $n=p$.