Prime ideal in residue class polynomial ring

Question

Consider the polynomial ring, $\mathbb{Z}[x,y]$ and the ideal $\mathfrak{a} = \langle x+2y\rangle$ in it. Let $\overline{a}, a \in \mathbb{Z}[x,y]$, represent the equivalence class, $a + \mathfrak{a}$.

Is $\langle \overline{x} \rangle$ a prime ideal in the residue class polynomial ring, $\mathbb{Z}[x,y]/\mathfrak{a}$? Since $\overline{x} = -\overline{2y}$, $\langle \overline{x} \rangle$ is not a prime ideal in $\mathbb{Z}[x,y]/\mathfrak{a}$?

Answer 1

Set $R=\mathbb{Z}[x,y]/\mathfrak{a}$. Then $R/\langle \overline{x} \rangle\simeq\mathbb{Z}[x,y]/\langle x,x+2y \rangle\simeq\mathbb Z[y]/\langle 2y \rangle$ which is not an integral domain for $\overline 2\overline y=\overline0$, and $\overline 2\ne\overline 0$, $\overline y\ne\overline 0$ in $\mathbb Z[y]/\langle 2y \rangle$.