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!

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 $X$.
