We have the ring $\Bbb Z_{13}$ and the polynomial $f(X,Y):=X-2Y \in \Bbb Z_{13}[X,Y]$.
We would like to prove two things:
1) The ideal $\langle X-2Y \rangle \trianglelefteq \Bbb Z_{13}[X,Y] $ is prime, but not maximal.
2) The polynomial $X-2Y $ is irreducible in $\Bbb Z_{13}[X,Y]$.
What are in general the teqniques one can follow, in order to show that a polynomial is irreducible in $K[X,Y]$, where $K$ is a field?
Answer. In the first one, it's a clasical teqnique to define the ring epimorphism $\phi:\Bbb Z_{13}[X,Y]\longrightarrow \Bbb Z_{13}[Y],\ f(X,Y)\longmapsto \phi(f(X,Y)):= f(2Y,Y)$ and then apply the 1st Isomorphism Theorem for Rings. So, it is $$\frac{\Bbb Z_{13}[X,Y]}{\langle X-2Y\rangle } \cong \Bbb Z_{13}[Y].$$ Then $\Bbb Z_{13}[Y]$ is an integral domain, but not a field. Equivalently, $\langle X-2Y \rangle$ is prime, but not maximal.
But any ideas for the other questions?
Idea. We know that if $R$ is an integral domain, $aT+b \in R[T]$ is irreducible in $R[T]$, if $a\in U(R)$ is invertible.
So, can we clame that it we take $f(X,Y)$ in $(\Bbb Z_{13}[Y])[X]=\Bbb Z_{13}[X,Y]$, then $X-2Y=1\cdot X-2Y \in (\Bbb Z_{13}[Y])[X]$ and $1\in U((\Bbb Z_{13}[Y])[X])=U(\Bbb Z_{13}[Y])=\Bbb Z_{13}^*$ and thus $f(X,Y)$ is irreducible in $(\Bbb Z_{13}[Y])[X]=\Bbb Z_{13}[X,Y]$?