Morphism from the Gaussian integers

#ring_theory

Let $i$ be the imaginary number, let $Z [i] = {a + b i ∣ a, b \in Z}$ , a principal ideal domain, and let $Z_{2}$ be the finite ring of integers modulo 2.
(a) Define a ring homomorphism from $Z [i] \to Z_{2}$ . You must prove it is a ring homomorphism.
(b) Find, with proof, a generator for the kernel of your ring homomorphism.

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Let $i$ be the imaginary number, let ${\bf Z}[i]=\{a+bi\,\mid \, a,b\in {\bf Z}\}$, a principal ideal domain, and let ${\bf Z}_2$ be the finite ring of integers modulo 2.
	\begin{enumerate}[label=\alph*)]
		\item Define a ring homomorphism from ${\bf Z}[i]\to {\bf Z}_2$. You must prove it is a ring homomorphism.
		\item Find, with proof, a generator for the kernel of your ring homomorphism.
	\end{enumerate}