Construct a bijection $Hom_{\mathbb C}(\mathbb C[x,y]/(xy-1),\mathbb C) \to \mathbb C - \{0\}$.
Here the similar question lies but I can not find the exact bijection which is explicitly defined.
I considered the mapping $\phi_{a,b} \mapsto {a \over b},$ where $a,b \in \mathbb C$ with $ab=1$ but I can not show it one-one. It is of course well-defined and surjective.
Can someone please say another mapping beyond it.
Thank you..