Proof that a certain domain is a valuation ring
Solution 1:
Your proof seems correct. If $y\neq 0$ take any $x/y\in \operatorname{Frac}A$. W.l.o.g. assume that $(y)\subset (x)$ in particular there is some $a\in A$ s.t. $xa=y$ and thus, $x/y= \frac{x}{xa} = \frac{1}{a} = (\frac{a}{1})^{-1}$ and because $a/1\in A$ the claim follows. One tiny thing to note ist that $\in A$ here means that the element is in the image of the canonical morphism $\phi : A\to \operatorname{Frac} A, \ x\mapsto x/1$ which establishes a canonical isomorphism $A \cong \phi(A) \subset \operatorname{Frac}A$.