Let $\mathbb{Z}[x]$ be the ring of polynomials in one variable over $\mathbb{Z}$.
I want to consider the quotient ring $R=\mathbb{Z}[x]/(2x)$. I would like to know how to find the prime ideals of $R$.
I am not sure how to tackle this question. I know about the classification of prime ideals of $\mathbb{Z}[x]$. However, I don't know how the ring $R$ looks like.
Does this even matter? Could we use the correspondence between prime ideals of the quotient ring $R$ and prime ideals of $\mathbb{Z}[x]$ containing $(2x)$?
Any hints are appreciated.