Correspondence Theorem for Ring Epimorphisms
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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 $B$.
Bijection
The direct image mapping $\pi^\to$ and the inverse image mapping $\pi^\gets$ induce reverse bijections between $I$ and $J$, specifically:
- For every ideal $\mathfrak a \in I$, its image $\pi^{\to}(\mathfrak a) = \pi(\mathfrak a) \in J$.
- For every ideal $\mathfrak b \in J$, its preimage $\pi^{\gets}(\mathfrak b) = \pi^{-1}(\mathfrak b) \in I$.
- 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 $A \overset \pi \to B \to B / \map \pi {\mathfrak a}$ 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 if and only if $\map \pi {\mathfrak a}$ is.
Maximal ideals
Let $\mathfrak a \in I$ be an ideal of $A$.
Then $\mathfrak a$ is a maximal ideal if and only if $\map \pi {\mathfrak a}$ is.
Closed embedding of prime spectrum
The induced map on spectra $\Spec \pi : \Spec B \to \Spec A$ is a topological closed embedding.