Just a theoretical question for now. without the exercise itself. Let $Z[x]$ be the ring and some ideal $I$. $I$ can be factored to (for example) 2 prime polynomials.
Are they any other zero divisors (none trivial), beside the prime polynomials which are the factorization of I?
As I understand the factored polynomials are defintly none-trivial zero divisors, because there multiply gives us I and modulo I it is 0, but none of them is 0.
The ring $ R=\mathbf Z_3[x]/(x^3+2x^2+2)$ has only two prime ideals $(x+1)R$ and $(x^2+2x+2)R$, since in $\mathbf Z_3[x]$, they are the irreducible factors of $x^3+2x^2+2$. Thus $ R$ is a semi-local ring of Krull dimension $0$. Its zero-divisors is the union of the two prime ideals in $\operatorname{Ass}R=\operatorname{Spec}R$.