Coordinate ring ($k[X]$)

For an affine algebraic set $X\subseteq {\mathbb{A}}_k^n$, its coordinate ring isThe set $X$ is irreducible exactly when $k[X]$ is an integral domain.

For an affine algebraic set $X$ over an algebraically closed field,If $fg$ vanishes on $X$, then $X$ is the union of its intersections with $V(f)$ and $V(g)$. Conversely, a nontrivial decomposition into closed subsets supplies functions vanishing on each proper piece whose product vanishes on all of $X$.

A morphism $f:X\to Y$ of affine varieties is dominant when its image is Zariski dense in $Y$. It is dominant exactly when the pullback homomorphismis injective. When $X$ and $Y$ are irreducible, this injection extends to their function fields and gives $\dim X\ge \dim Y$.