This question have two parts. I was able to do the first part.
Consider the ring
$R = \{(a, b) \in \mathbb{Z} \times \mathbb{Z} | \, a \equiv b\, \mod 5 \}$
a) Show that the homomorphism $f: \mathbb{Z}[x] \rightarrow R $ sending $1$ to $(1, 1)$ and $x$ to $(5, 0)$ is surjective with kernel $(x^2 − 5x)$.
This part was fairly easy to show. I assume it is related to the second part
b) Find all prime ideals of $R$ containing $f(3)= 3 \cdot 1_{R}$. Then find all prime ideals of $R$ containing $f(5)= 5 \cdot 1_{R}$.
For this part I am trying to use fact that pre-images of prime ideals under homeomorphism of rings are prime ideal as well. Hence pre-image of prime ideal of $R$ containing $f(3)= 3 \cdot 1_{R}$ has tobe prime ideal containing ideal $(3)\subset \mathbb{Z}[x]$. It also has to contain kernel $(x^2-5x)$.
Now part I am not sure about: since $f$ is homeomorphism we have isomorphism $\mathbb{Z}[x]/(x^2-5x) \cong R$. I can show that no polynomial of degree less than $2$ is mapped to $3\cdot 1_{R}$ except for constant polynomial $h(x)=3$.
Would this somehow imply that pre-image of prime ideal containing $3\cdot 1_{R}$ is necessary $(x^2-5x, 3)$?
If this is correct, can I use same construction for $5 \cdot 1_{R}$ ?