Correspondence Theorem for Ring Epimorphisms

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

Let $\pi : A \to B$ be a ring epimorphism.

Let $I$ be the set of ideals of $A$ containing the kernel $\operatorname{ker} f$.

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

Bijection
The direct image mapping $\pi^\to$ and the inverse image mapping $\pi^\gets$ induce reverse bijections between $I$ and $J$, specifically:


 * 1) For every ideal $\mathfrak b \in I$, its image $\pi^{\to}(\mathfrak b) = \pi(\mathfrak b) \in J$.
 * 2) For every ideal $\mathfrak c \in J$, its preimage $\pi^{\gets}(\mathfrak c) = \pi^{-1}(\mathfrak c) \in I$.
 * 3) The restrictions $\pi^\to : I \to J$ and $\pi^\gets : J \to I$ are reverse bijections.

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

Also see

 * Correspondence Theorem for Quotient Rings
 * Correspondence Theorem for Surjective Module Homomorphisms