Let $n$ be an integer and $f_{i}$, $i \in I$, a finite collection of integer polynomials.
Let $(n,f_i)$ be the ideal generated in the ring $\mathbb{Z}[x]$ by these elements. If $(n)=n\mathbb{Z}[x]$, the ideal generated by $n$ inside $\mathbb{Z}[x]$, then $(n)\subset (n,f_i)$, so we can apply the "third" isomorphism theorem for rings (I understand some call it the "second" isomorphism theorem) to get: $$ \mathbb{Z}[x]\Big/(n,f_i)\cong\big(\mathbb{Z}[x]/(n)\big)\Big/\big((n,f_i)/(n)\big)\cong\mathbb{Z}_n[x]\Big/(\bar{f_i}) $$ where $\bar{f_i}$ are the images of $f_i$, under the (natural) projection $\mathbb{Z}[x]\twoheadrightarrow\mathbb{Z}[x]/(n)\cong\mathbb{Z}_n[x]$.
Is this correct? Is there some other way around, to prove the isomorphism $$ \mathbb{Z}[x]\Big/(n,f_i)\cong\mathbb{Z}_n[x]\Big/(\bar{f_i})\ ? $$ Edit: In fact, the motivation for asking this, is that I was attending a lecture the other day, where someone from the audience claimed that there is number-theoretic interpretation of such isomorphisms, but I am not sure what he was talking about (the topic was quite different).