Irreducibility and prime vanishing ideal
= Irreducibility and prime vanishing ideal
For an affine algebraic set $X$ over an algebraically closed field,
$$
X\text{ is irreducible}
\quad\Longleftrightarrow\quad
I(X)\text{ is a prime ideal}.
$$
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 $X$.