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 $\ker \pi$.

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 $A \overset \pi \to B \to B / \map \pi {\mathfrak a}$ with the quotient ring epimorphism has kernel $\mathfrak a$ and there is an isomorphism $A / \mathfrak a \to B / \map \pi {\mathfrak a}$.

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

Then $\mathfrak a$ is a prime ideal $\map \pi {\mathfrak a}$ is.

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

Then $\mathfrak a$ is a maximal ideal $\map \pi {\mathfrak a}$ is.

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

Also see

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