The structure of the integers as both a group and a ring

Consider the additive group of integers ${\bf Z}$.
\begin{enumerate}[label=\alph*)]
	\item Prove that every subgroup of ${\bf Z}$ is a cyclic group.
	\item Prove that every homomorphic image of ${\bf Z}$ is a cyclic group.
	\item Now consider the {\itshape ring} ${\bf Z}$. Exhibit a prime ideal of ${\bf Z}$ that is not maximal.
\end{enumerate}