The question is about the proof that ${\mathbb{Z}\left[ i \right] /(i+2)}$ is isomorphic to $\mathbb{Z} /p\mathbb{Z}$.
To prove this problem,We restrict homomorphism $R \to R/I$ to $\mathbb{Z} \to R/I$ in order to show that image of R is the same as image of $\mathbb{Z}$.
I know the kernel of the restricted hom is $5\mathbb{Z}$ but I don't understand why the image of the restricted hom is $\mathbb{Z} /5\mathbb{Z}$ instead of $\mathbb{Z}/I$?
I was confused by the mod $(a+bi)$ and $p\mathbb{Z}$ in the proof.
