Proposition 11

*Let $A$ be a ring, $a$ an ideal of $A$ and $Y$ a subset of $X = \mathrm{Spec}(A)$. (i) $V(a)$ is closed in $X$ and $\mathfrak{J}(Y)$ is an ideal of $A$ which is equal to its radical. (ii) $\mathfrak{J}(V(a))$ is the radical of $a$ and $V(\mathfrak{J}(Y))$ is the closure of $Y$ in $X$.*

(iii) The mappings 3 and V define decreasing bijections, one of which is the inverse of the other, between the set of closed subsets of X and the set of ideals of A which are equal to their radicals.

Assertion (i) and the first assertion of (ii) follow from the definitions and § 2, no. 6, Corollary 1 to Proposition 13. If a closed set V(M) (for some M ⊂ A) contains Y, then M ⊂ p for every prime ideal p ∈ Y, whence M ⊂ J(Y) and consequently V(M) ⊃ V(J(Y)); as Y ⊂ V(J(Y)), V(J(Y)) is the smallest closed set of X containing Y, which completes the proof of (ii). Finally, it follows from (ii) that, if a is a prime ideal equal to its radical, then J(V(a)) = a and that, if Y is closed in X, then V(J(Y)) = Y; this proves (iii).

It follows immediately from Proposition 11 that, if M is any subset of A and Y is any subset of X, then V(M) = V(J(V(M))) and J(Y) = J(V(J(Y))).