Prime ideals of $R= \mathbb{Z}_{(2)}[x]/(2x)$

Question

I want to prove $\dim R=1$. So I think I have to prove that R has only 1 prime ideal. That's why I ask this question

Answer 1

Prime ideals of a quotient $R/I$ are prime ideals of $R$, that contain $I$.

$P \supset (2x)$ is equivalent to $2 \in P$ or $x \in P$.

We have $R/(2) = \mathbb F_2[x]$ and $R/(x)= \mathbb Z_{(2)}$, hence the prime ideals are given by

$$(2),(x) ~ \text{ and } ~ (2,f)$$ where $f \in \mathbb Z[X]$ is irreducible modulo $2$. In particular there are infinitely many of them. Though the dimension is $1$, because all chains are of the form

$$(2) \subset (2,f) ~ \text{ or } ~ (x) \subset (2,x).$$