Let \(g(X)\) be the minimal poly of \(\gamma\). Then we have
\[ \mathbb{Z}[X]/\langle g(X) \rangle \cong \mathbb{Z}[\gamma] \]
So then we can see that
\[ \mathbb{Z}[\gamma]/\langle p \rangle \cong \mathbb{Z}[X] / \langle g(X), p \rangle \]

My question: is this a general rule of rings that given a homomorphism
\[ \phi : R/I \rightarrow S \]
with ker \(\phi = J\), then
\[ R/(I + J) \cong S/J \]
?