Coordinate ring

Remarks

Two polynomials have the same class in $k[X]$ if and only if they define the same function on $X$. So $k[X]$ is the ring of polynomial functions $X \to k$. It is a finitely generated $k$-algebra, generated by the classes of $x_1, \dots, x_n$. It is a reduced ring, since $I(X)$ is a radical ideal.

Example: for the parabola $X = V(y - x^2)$ over an infinite field, $k[X] = k[x, y]/(y - x^2) \cong k[x]$, through $x \mapsto x$ and $y \mapsto x^2$.

Non-example: the ring $k[x]/(x^2)$ is not the coordinate ring of any algebraic set, because it has the nonzero nilpotent element $x$.

$X$ is irreducible if and only if $k[X]$ is an integral domain; when $k$ is algebraically closed, this says that $X$ is an affine variety. If $k$ is algebraically closed, Hilbert's Nullstellensatz shows that the points of $X$ correspond to the maximal ideals of $k[X]$. The point $a$ corresponds to the ideal of functions that vanish at $a$.