The Correspondence Theorem for Rings

The correspondence theorem establishes a lattice isomorphism between ideals of a quotient ring and ideals of the original ring containing the kernel, providing complete structural information about quotient rings.

Statement

Theorem5.6Correspondence theorem (fourth isomorphism theorem)

Let $I$ be an ideal of a ring $R$ and $\pi: R \to R/I$ the canonical projection. There is an inclusion-preserving bijection:

$\Phi: \{J : J \text{ ideal of } R,\, I \subseteq J\} \xrightarrow{\sim} \{\bar{J} : \bar{J} \text{ ideal of } R/I\}$

given by $\Phi(J) = J/I = \pi(J)$ , with inverse $\Phi^{-1}(\bar{J}) = \pi^{-1}(\bar{J})$ . This bijection preserves:

Sums: $\Phi(J_1 + J_2) = \Phi(J_1) + \Phi(J_2)$ .
Intersections: $\Phi(J_1 \cap J_2) = \Phi(J_1) \cap \Phi(J_2)$ .
Products: $\Phi(J_1 J_2) = \Phi(J_1)\Phi(J_2)$ .
Primality and maximality.

Proof

$\Phi$ is well-defined: If $I \subseteq J$ and $J$ is an ideal, then $J/I = \{j + I : j \in J\}$ is an ideal of $R/I$ (if $j_1, j_2 \in J$ and $r \in R$ : $(j_1+I)-(j_2+I) = (j_1-j_2)+I \in J/I$ and $(r+I)(j_1+I) = rj_1+I \in J/I$ ).

$\Phi^{-1}$ is well-defined: If $\bar{J}$ is an ideal of $R/I$ , then $\pi^{-1}(\bar{J}) = \{r \in R : r + I \in \bar{J}\}$ is an ideal of $R$ containing $I$ .

$\Phi$ and $\Phi^{-1}$ are inverses: $\Phi(\Phi^{-1}(\bar{J})) = \pi(\pi^{-1}(\bar{J})) = \bar{J}$ (since $\pi$ is surjective). $\Phi^{-1}(\Phi(J)) = \pi^{-1}(\pi(J)) = J$ (since $I \subseteq J$ ensures $\pi^{-1}(\pi(J)) = J$ ).

Preserves primality: $J/I$ is prime in $R/I$ iff $(R/I)/(J/I) \cong R/J$ is an integral domain iff $J$ is prime in $R$ .

Preserves maximality: Similarly, $J/I$ is maximal iff $R/J$ is a field iff $J$ is maximal. $\blacksquare$

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Applications

ExampleStructure of quotient rings

Ideals of $\mathbb{Z}/12\mathbb{Z}$ : Divisors of 12 give ideals $(1), (2), (3), (4), (6), (12)$ in $\mathbb{Z}$ . Prime ideals: $(\bar{2}), (\bar{3})$ . These are also the maximal ideals.
Ideals of $k[x]/(x^3 - x)$ : Factor $x^3 - x = x(x-1)(x+1)$ . Ideals containing $(x^3-x)$ are $(x), (x-1), (x+1), (x(x-1)), (x(x+1)), ((x-1)(x+1)), (x^3-x), (1)$ . By CRT: $k[x]/(x^3-x) \cong k \times k \times k$ .
Chains of primes: The correspondence preserves chains, so $\dim(R/I) = \dim(R) - \mathrm{ht}(I)$ in well-behaved rings (the dimension formula for Noetherian rings).

RemarkLattice structure

The correspondence is an isomorphism of lattices: it preserves the meet ( $\cap$ ) and join ( $+$ ) operations. This means the ideal structure of $R/I$ is completely determined by the ideal structure of $R$ "above" $I$ . In algebraic geometry, this translates to: the subvarieties of $V(I)$ correspond to the ideals containing $I$ .