Isomorphism Gaussian and Polynomial Quotients How does $$\mathbb{Z}[i]/(2+i) \cong \mathbb{Z}[x]/(x^2+1,2+x)$$ follow formally from the fact that $$\mathbb{Z}[i] \cong \mathbb{Z}[x]/(x^2+1)?$$

$\newcommand{\ZZ}{{\mathbb Z}}$ 
$\newcommand{\ideal}[1]{{\mathfrak #1}}$
It can be transformed
$$
\begin{multline}
\ZZ[i]/(2+i)\ZZ[i] = (\ZZ[x]/(x^2+1))/((2+\bar{x})\ZZ[x]/(x^2+1)) =  \\ =(\ZZ[x]/(x^2+1))/((2+x) \ZZ[x]/(x^2+1)) = (\ZZ[x]/(x^2+1))/((2+x,x^2+1)/(x^2+1)) = \\
 = \ZZ[x]/(2+x,x^2+1)
\end{multline}
$$
using $(M/P)/(N/P) = M/N$ for $A$-modules $M \supseteq N \supseteq P$ (Noether isomorphism theorem) and $\ideal{a} (A/\ideal{b}) = (\ideal{a} + \ideal{b})/\ideal{b}$ for $\ideal{a}, \ideal{b} \subseteq A$ two ideals of $A$.
