Isomorphism in coordinate ring Let $x_{1},x_{2},...,x_{m}$ be elements of $\mathbb{A}^{n}$, where $\mathbb{A}^{n}$ is the n-affine space over an algebraically closed field $k$. Now define $X=\{x_{1},x_{2},...,x_{m}\}$. Why is the coordinate ring $A(X)$, isomorphic to $\oplus_{j=1}^{m} k = k^{m}$?

For each $i$, $A(\{x_i\}) = k[x]/I(x_i) \cong k$, so each $I(x_i)$ is a maximal ideal of
$k[x]$.  I assume the points $x_1,\ldots,x_n$ are distinct, from which it follows easily
that the ideals $I(x_1),\ldots,I(x_n)$ are distinct maximal ideals.  Thus they are pairwise comaximal and the Chinese Remainder Theorem -- see e.g. $\S 4.3$ of these notes -- applies.  I leave it to you to check that it gives the conclusion you want.
