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 $B$.

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

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 if and only if $\pi(\mathfrak a)$ is.

Maximal ideals

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

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