Correspondence Theorem for Quotient Rings/Bijection
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Theorem
Let $A$ be a commutative ring with unity.
Let $\mathfrak a \subseteq A$ be an ideal.
Let $A / \mathfrak a$ be the quotient ring and $\pi : A \to A / \mathfrak a$ the quotient ring epimorphism.
The direct image mapping $\pi^\to$ and the inverse image mapping $\pi^\gets$ induce reverse bijections between the ideals of $A$ containing $\mathfrak a$ and the ideals of $A/\mathfrak a$, specifically:
Let $I$ be the set of ideals of $A$ containing $\mathfrak a$.
Let $J$ be the set of ideals of $A / \mathfrak a$.
Then:
- $(1): \quad$ For every ideal $\mathfrak b \in I$, its image $\map {\pi^\to} {\mathfrak b} = \map \pi {\mathfrak b} \in J$
- $(2): \quad$ For every ideal $\mathfrak c \in J$, its preimage $\map {\pi^\gets} {\mathfrak c} = \map {\pi^{-1} } {\mathfrak c} \in I$
- $(3): \quad$ The restrictions $\pi^\to: I \to J$ and $\pi^\gets: J \to I$ are reverse bijections.
Proof
Follows from Correspondence Theorem for Ring Epimorphisms/Bijection
$\blacksquare$