The kernel of an evaluation morphism

#ring_theory

Let $Z [X]$ be the ring of polynomials with integer coefficients, and let $K \subset Z [X]$ be the kernel of the "evaluation at $1$ " homomorphism

\begin{aligned} ε_{1} : Z [X] & \to Z_{3} \\ f (X) & \mapsto [f (1)]_{3} . \end{aligned}

(a) Characterize $K$ as a set.
(b) Determine whether $K$ is a maximal ideal. Fully justify your conclusion.
(c) Determine whether $K$ is a principal ideal. Justify by either exhibiting a generator or proving that there isn't one.

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Let ${\bf Z}[X]$ be the ring of polynmomials with integer coefficients, and let $K\subset {\bf Z}[X]$ be the kernel of the ``evaluation at $1' homomorphism
\begin{align*}
	\varepsilon_1:{\bf Z}[X] &\to {\bf Z}_3\\
	f(X) &\mapsto [f(1)]_3.
\end{align*}
\begin{enumerate}[label=\alph*)]
	\item Characterize $K$ as a set.
	\item Determine whether $K$ is a maximal ideal. Fully justify your conclusion.
	\item Determine whether $K$ is a principal ideal. Justify by either exhibiting a generator or proving that there isn't one.
\end{enumerate}