Corollary 2

The mapping $p \mapsto V(p)$ is a bijection of $X = \mathrm{Spec}(A)$ onto the set of irreducible closed subsets of $X$; in particular the irreducible components of a closed subset $Y$ of $X$ are the sets $V(p)$, where $p$ runs through the set of minimal elements of the set of prime ideals of $A$ which contain $\mathfrak{J}(Y)$.

As $\mathfrak{J}(V(p)) = p$ for every prime ideal $p$ of $A$ and $Y = V(\mathfrak{J}(Y))$ for every closed subset $Y$ of $X$, the first assertion follows from Proposition 14; on the other hand, for $Y \supset V(p)$, it is necessary and sufficient that

(Proposition 11), whence the second assertion.