# Correspondence Theorem for Quotient Rings

Jump to navigation
Jump to search

## 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:

- $(1): \quad$ 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.

A particular theorem is missing. In particular: inclusion-preserving, prime ideals, maximal ideals, intermediate quotients, closed embedding, ... |

## Also see

- First Isomorphism Theorem for Rings
- Universal Property of Quotient Ring
- Correspondence Theorem for Localizations of Rings