I have a three-part problem I'm working on, and the first and last part are easy, but the second is giving me trouble.
Let $n\in\mathbb{Z}$ with $\sqrt{n}\notin\mathbb{Z}$, and consider the ring $R=\mathbb{Z}[\sqrt{n}]=\{a+b\sqrt{n}\,|\,a,b\in\mathbb{Z}\}$. Prove:
(a) for every non-zero $z\in R$, there is a non-zero z'\in R such that zz'\in\mathbb{Z}.
(b) Conclude from (a) that $R/I$ is finite for every non-zero ideal $I$.
(c) Conclude from (b) that every non-zero prime ideal of $R$ is maximal.
Now, part (a) is very straightforward: take $z=a+b\sqrt{n}$ non-zero in $R$. Then $a$ and $b$ are not both zero. So z'=a-b\sqrt{n} is also non-zero in $R$. And zz'=a^2-b^2n\in\mathbb{Z}.
Further, given that we believe (b), part (c) is also easy: let $I$ be a non-zero prime ideal in $R$. Then $R/I$ is a finite integral domain, and thus a field. So in fact $I$ is a maximal ideal.
However, I'm having trouble with (b). What I have so far: Let $I$ be a non-zero ideal of $R$ and suppose $R/I$ is not finite. Take $z\neq 0$ in $I$. Then there is some z'\neq 0 in $R$ such that zz'\in\mathbb{Z}. But $I$ is an ideal, so in fact zz'\in I....
And here's where I've stalled. Any help is appreciated. Thanks!