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 $\operatorname{ker} \pi$.

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

Inclusion-preserving
The mappings $\pi^\to : I \to J$ and $\pi^\gets : J \to I$ are inclusion-preserving.

Isomorphism between Quotient Rings
Let $\mathfrak a \in I$ be an ideal of $A$.

Then the composition $A \overset \pi \to B \to B/\pi(\mathfrak a)$ with the quotient ring epimorphism has kernel $\mathfrak a$ and there is an isomorphism $A/\mathfrak a \to B/\pi(\mathfrak a)$.

Prime ideals
Let $\mathfrak a \in I$ be an ideal of $A$.

Then $\mathfrak a$ is a prime ideal $\pi(\mathfrak a)$ is.

Maximal ideals
Let $\mathfrak a \in I$ be an ideal of $A$.

Then $\mathfrak a$ is a maximal ideal $\pi(\mathfrak a)$ is.

Closed embedding of prime spectrum
The induced map on spectra $\operatorname{Spec} \pi : \operatorname{Spec} B \to \operatorname{Spec} A$ is a topological closed embedding.

Also see

 * Correspondence Theorem for Quotient Rings
 * Correspondence Theorem for Module Epimorphisms