Correspondence Theorem for Quotient Rings

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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.


Bijection

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




Also see


Generalization