Prove that $R/(2i)$ and $\mathbb{Z}/4\mathbb{Z}$ are isomorphic rings Let $R=\{a+2ib|a,b\text{ integers}\}$.Prove that $R/(2i)$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}$

Often, when trying to show that there exists an isomorphism between a quotient ring $R/I$ and another ring $S$, it is easiest to find a surjective homomorphism $R\to S$ which has kernel $I$, and then to use the isomorphism theorem.
In this case, consider the map
$$\begin{align}R &\to \mathbb Z/4\mathbb Z\\a+2ib&\mapsto a\pmod 4\end{align}$$
Can you show that this:


*

*is a homormorphism

*is surjective

*has kernel $(2i)\subset R$

