Correspondence Theorem for Ring Epimorphisms

Theorem
Let $A$ and $B$ be commutative rings with unity.

Let $f : A \to B$ be a ring homomorphism.

Let $K \subset A$ be its kernel.

Let $S$ be the set of ideals of $A$ containing $K$.

Let $T$ be the set of ideals of $B$.

Let $f^\to : \mathcal P(A) \to \mathcal P(B)$ be the direct image mapping of $f$.

Let $f^\gets : \mathcal P(B) \to \mathcal P(A)$ be the inverse image mapping of $f$.

Then:


 * 1) For all $I \in S$, $f^\to(I) \in T$.
 * 2) For all $J \in T$, $f^\gets(J) \in S$.
 * 3) $f^\to : S \to T$ is an inclusion-preserving bijection whose inverse is $f^\gets$.

Proof
The first statements follow from:
 * Preimage of Ideal under Ring Epimorphism is Ideal
 * Image of Ideal under Ring Epimorphism is Ideal

Also see

 * Correspondence Theorem for Surjective Module Homomorphisms