TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-23 (gpt-6-sol)

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The radicals in a minimal primary decomposition are intrinsic

Statement

Assume the Axiom of Choice.

Let $R$ be a Noetherian commutative ring, let $M$ be a finitely generated left $R$ -module, and let $N \subseteq M$ . Let $N = Q_{1} \cap \dots \cap Q_{r}$ be a minimal primary decomposition in which each $Q_{i}$ is $p_{i}$ -primary for a prime ideal $p_{i}$ . Then the set of component radicals is uniquely determined by $M / N$ and equals ${Ass}_{R} (M / N) .$

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice); a Noetherian commutative ring $R$ , a finitely generated left $R$ -module $M$ , a submodule $N \subseteq M$ , and a minimal primary decomposition $N = Q_{1} \cap \dots \cap Q_{r}$ with each $Q_{i}$ $p_{i}$ -primary for a prime ideal $p_{i}$ .

[L1]

Under the Axiom of Choice, in the Noetherian finite-module setting, if a minimal primary decomposition has each component $p_{i}$ -primary for a prime ideal $p_{i}$ , then its radicals are exactly the associated primes of the quotient (The radicals in a minimal primary decomposition are exactly the associated primes of the quotient).

Proof

technique · direct

1.1L1given

For the given minimal primary decomposition, [L1] gives ${p_{1}, \dots, p_{r}} = {Ass}_{R} (M / N) .$

2.1step 1.1∎

The right-hand side depends only on the quotient $M / N$ , not on the chosen decomposition. Hence the set of component radicals is intrinsic.

Depends on

Used by

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

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Theorem (18.20) (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, v4.03, Theorem 19.10 (standard reference, not scraped)