Correspondence Theorem for Ring Epimorphisms/Bijection

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 $B$.

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.

Proof

The first statements follow from:

• Preimage of Ideal under Ring Homomorphism is Ideal
• Image of Ideal under Ring Epimorphism is Ideal

The last statement follows from:

• Image of Preimage of Ideal under Ring Epimorphism
• Preimage of Image of Ideal under Ring Homomorphism

$\blacksquare$