LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-27

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Radicals and quotient correspondence

Statement

Let $R$ be a commutative ring, let $I \subseteq J ⊴ R$ be ideals, and write $π : R \to R / I$ for the quotient map. Then $\sqrt{J / I} = \sqrt{J} / I$ as ideals of $R / I$ . In particular, $J / I$ is radical in $R / I$ if and only if $J$ is radical in $R$ .

Facts & Assumptions

Given: A commutative ring $R$ , ideals $I \subseteq J ⊴ R$ , and the quotient map $π : R \to R / I$ .

[L1]

Ideals of $R / I$ correspond to ideals of $R$ containing $I$ , so $J / I$ is an ideal of $R / I$ (Correspondence theorem: ideals of $R / I$ correspond to ideals of $R$ containing $I$ ).

[L2]

An element belongs to the radical of an ideal exactly when one of its positive powers lies in that ideal (The radical of an ideal).

Proof

technique · direct

1.1L1L2givenalgebra

Let $x + I \in R / I$ . By [L2], $x + I \in \sqrt{J / I}$ exactly when $(x + I)^{n} = x^{n} + I \in J / I$ for some $n \geq 1$ , and that happens exactly when $x^{n} \in J$ . Applying [L2] again shows that this is equivalent to $x \in \sqrt{J}$ , so $x + I \in \sqrt{J / I}$ exactly when $x + I \in \sqrt{J} / I$ .

2.1step 1.1L1algebra

Step 1.1 proves $\sqrt{J / I} = \sqrt{J} / I$ . Consequently, $J / I$ is radical exactly when $J / I = \sqrt{J / I}$ , exactly when $J / I = \sqrt{J} / I$ , and exactly when $J = \sqrt{J}$ .

3.1step 1.1step 2.1∎

The quotient radical is therefore exactly the quotient of the radical, and radical ideals correspond across the quotient map.

Depends on

Used by

Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals Theorem

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

M. Hochster, Introduction to Commutative Algebra, Math 614 notes (2020) (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, v4.03, §2 Ideals (standard reference, not scraped)