Elementary question to Find the coordinate ring Kindly asking for any hints about the following questions: 
Suppose $k$ is an algebraically closed field of characteristic zero. If $X$ is the variety 
$k \setminus ‎\lbrace‎ 0, 1‎\rbrace‎$, then proof that $A \cong k[t, t^{−1}, (t−1)^{−1}]$, where $A$ is the coordinate ring of $X$.
Thanks for your help!
 A: As Cocopuffs suggested, consider the map $f\colon X \to \mathbb A^2$ defined by $x \mapsto \left(x, \frac{1}{x(x-1)}\right)$.
Then show the following:


*

*The image of $f$ is the zero set of $xy(x-1) - 1$.

*The map $f$ is an isomorphism onto its image.

*The coordinate ring, $B = \frac{k[x, y]}{xy(x-1) - 1}$ of the image is isomorphic to the ring $k[t, t^{-1}, (t - 1)^{-1}]$.


Then (1) shows that the image is a variety.  (2) shows that this variety is isomorphic to $X$ so the coordinate rings are equal, i.e., $k[X] \simeq B$.  Finally (3) shows that $B$ is the ring you were looking for all along.
A: Somewhat naïvely, the coordinate ring of the variety is the largest ring of rational functions you can define on $X$.
In this case, $k\backslash \{0,1\}$, the only rational functions you are allowed to include in your ring are those having only $0$ or $1$ as poles, because only these will be well-defined on $X$.
Thus your coordinate ring is $k[t,\frac{1}{t}, \frac{1}{1-t}]$, as this is the largest rational function ring you can define on $X$.
