Affine variety

Remarks

An algebraic set $X$ is irreducible if and only if its ideal $I(X)$ is a prime ideal. For instance, if $X$ is irreducible and $fg \in I(X)$, then $X = (X \cap V(f)) \cup (X \cap V(g))$, and irreducibility forces $f \in I(X)$ or $g \in I(X)$. With Hilbert's Nullstellensatz, affine varieties in $k^n$ correspond to prime ideals of $k[x_1, \dots, x_n]$.

Example: the parabola $X = V(y - x^2) \subseteq k^2$ is an affine variety. Its ideal is $(y - x^2)$, and $k[x, y]/(y - x^2) \cong k[x]$ is an integral domain, so the ideal is prime.

Non-example: $V(xy) \subseteq k^2$ is an algebraic set but not an affine variety. It is the union of the two axes, each strictly smaller than $V(xy)$.

Many texts, especially those based on schemes, call any algebraic set an affine variety, or use the term for a variety isomorphic to a closed subvariety of some $k^n$.