I got stuck at the following exercise:
Let $V$ be an (unitary) integral domain contained in a field $K$. Then $V$ is a valuation ring with quotient field $K$ if and only if for every $u \in K$ either $u \in V$ or $u^{-1} \in V$.
Here is the definition of valuation ring that I know:
Definition: A commutative ring $V$ is called valuation ring if for all $a, b \in V$ holds either $a \mid b$ or $b \mid a$.
Here is what I got so far:
$\Longrightarrow$: Assume that for an $u \in K$ holds both $u \in V$ and $u^{-1} \in V$. Since $1 = u u^{-1}$ this implies that $u \mid 1$. But this is a contradiction to our assumption that $V$ is a valuation ring, since it (trivially) also holds that $1 \mid u$.
$\Longleftarrow$: This is the direction where I got stuck. I tried a proof by contradiction, so assume for some $a,b \in V$ holds $a \mid b$ and $b \mid a$, but I do not know how to continue.
Could you give me a hint?
You might like to look at the article "On Valuation Rings" which I wrote with a colleague a little time ago. You can find it on the web. RC