Is $\mathbb{Z}[x]/(5x)$ isomorphic to $\mathbb{F}_5[x]$?

Question

It seems to me that $\mathbb{Z}[x]/(5)$ is isomorphic to $\mathbb{F}_5[x]$, but $\mathbb{Z}[x]/(5x)$ isn't? Since, for example, $x+5 \neq x \in \mathbb{Z}[x]/(5x)$ but they are equal in $\mathbb{F}_5[x]$.

Is this reasoning correct? If so, how can I formalize it?

Answer 1

In any ring $R$, if you have elements $a,b\in R$ such that $(a)\neq (ab)$ and $(b)\neq (ab)$, then obviously $R/(ab)$ cannot be a domain since it has nonzero zero divisors $a+(ab)$ and $b+(ab)$.

In particular, this is the case when $R$ is a domain and $a,b$ are two nonzero elements, as in your case.

Answer 2

Yes, you are right. $\mathbb{Z}[x]/(5x)$ is isomorphic to the subring of product ring $\mathbb{F}_5[x]\times\mathbb{Z}$ of pairs $(f(x),n)$ such that $f(0)\equiv 0 \pmod5$