TheoremStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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Prime ideals of a quotient ring are exactly the prime ideals containing the ideal

Statement

Let $R$ be a commutative ring, let $I ⊴ R$ be an ideal, and let $π : R \to R / I$ be the quotient map. Then contraction along $π$ induces an inclusion-preserving bijection $Spec (R / I) \to V (I)$ , sending $q$ to $π^{- 1} (q)$ . Its inverse sends a prime ideal $p \supseteq I$ to $p / I$ .

Facts & Assumptions

Given: A commutative ring $R$ , an ideal $I ⊴ R$ , and the quotient map $π : R \to R / I$ .

[L1]

Primes of $R / I$ correspond to primes of $R$ containing $I$ , and strict inclusions are preserved (Primes of a quotient lie over the kernel).

[L2]

Every quotient map induces a spectrum map by contraction (A ring map induces a contraction map on prime spectra).

Proof

technique · direct

1.1

By [L2], contraction along $π$ gives a map $Spec (R / I) \to Spec (R)$ . The quotient-prime correspondence [L1] shows that its values are precisely prime ideals containing $I$ , so the map lands in $V (I)$ .

L1L2

1.2

The same correspondence [L1] provides the inverse assignment $p \mapsto p / I$ on $V (I)$ , and it also shows that extension and contraction undo one another and preserve inclusion.

2.1

Therefore contraction along $π$ identifies $Spec (R / I)$ with $V (I)$ .

step 1.1step 1.2∎

Depends on

Used by

Dimension of a quotient via chains above an ideal Corollary
Passing to the reduced quotient does not change the prime spectrum Corollary
Dual numbers and their reduced quotient have the same prime set Example
Minimal and maximal primes of the node ring Example

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §13 and §17 (standard reference, not scraped)