Every quotient of a number ring is finite Let $K$ be a number field, i.e. a subfield of $\mathbb{C}$ of finite degree over $\mathbb{Q}$. Let $\mathscr{O}_K$ be the ring of integers of $K$, i.e. algebraic integers which are in $K$. Let $I$ be an ideal of $\mathscr{O}_K$. I read many times that the quotient $\mathscr{O}_K/I$ is obviously/clearly a finite ring, but i've never seen a proof. Could someone suggest me how to see this?

I guess you know that, as an Abelian group, $O_K\cong\mathbb Z^k$, where $k=[K:\mathbb Q]$. Now if $0\neq a\in J$ then $(a)\subset J$, and as $a$ divides its norm $Na\in\mathbb Z$, also $(Na)\subset(a)$. And $\mathbb Z^k/(Na)\mathbb Z^k$ is certainly finite.
