I am reading Eisenbud and Harris's The Geometry of Schemes. Exercise I-20 in it is to calculate the points and sheaf of functions for some schemes.
$1)$ $X=$Spec $\mathbb C[x]/(x^{2}-x)$. We know that it only has two closed points: $(x),(x-1)$. Call them $a,b$. The topology then should be$\{\emptyset,\{a\},\{b\},\{a,b\}\}$. We have that $\mathcal O(\emptyset)=0$,$\mathcal O(\{a,b\})=\mathbb C[x]/(x^{2}-x)$. Now I want to calculate $\mathcal O(\{a\})$
Note that $\{a\}=\{(x)\}=X_{x-1}, X_{x-1}$ is the basic open subset of $X$. So $\mathcal O(\{a\})=\mathcal O(X_{x-1})=(\mathbb C[x]/(x^2-x))_{x-1}$.
Am I right? If I am right, how to simplify $(\mathbb C[x]/(x^2-x))_{x-1}$