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

#ring_theory

Consider the additive group of integers $Z$ .
(a) Prove that every subgroup of $Z$ is a cyclic group.
(b) Prove that every homomorphic image of $Z$ is a cyclic group.
(c) Now consider the ring $Z$ . Exhibit a prime ideal of $Z$ that is not maximal.

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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}