(a) Give a specific example of a PID with exactly two prime ideals. Give a brief proof of your answer.
(b) Give an specific example of a PID with infinitely many prime ideals. Give a brief proof of your answer.
My ideas:
(a) $\mathbb Z_{(p)}$: localize $\mathbb Z$ at prime ideal p so $0_{(p)} \subset (p)_{(p)}$
A PID has $0$ and one other maximal ideal so it is a quasilocal PID
Is this a correct example and how should I 'give a brief proof'??
(b) $\mathbb Z$: only has $1$ and $-1$ as units and PID's are $1$-dimensional so only $0$ and minimal ideal exist
Proof Idea: Suppose $p_1...p_n$ are all positive primes. Then $p_1...p_{n+1}$ is not divisible by any prime $p_i$ so it is a unit which is a contradiction.
Any feedback or suggestions would be appreciated! Thanks.