LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Parameter ideals are exactly the m-primary d-generated ideals

Statement

Let $(R, m)$ be a $d$ -dimensional Noetherian local ring and let $J = (x_{1}, \dots, x_{d})$ . Then $(x_{1}, \dots, x_{d})$ is a system of parameters if and only if $J$ is $m$ -primary.

Facts & Assumptions

Given: A $d$ -dimensional Noetherian local ring $(R, m)$ and the ideal $J = (x_{1}, \dots, x_{d})$ .

[L1]

By definition, $(x_{1}, \dots, x_{d})$ is a system of parameters exactly when $\sqrt{J} = m$ (Systems of parameters and parameter ideals).

[L2]

For a finite module over a Noetherian ring, a proper ideal is $p$ -primary exactly when the quotient has associated-prime set ${p}$ ; equivalently, when some power of $p$ kills the quotient and every element outside $p$ acts injectively (Primary submodules of finite modules are characterized by a singleton associated-prime set).

Proof

technique · direct

1.1

Suppose $\sqrt{J} = m$ . Since every element outside $m$ is a unit in a local ring, it acts injectively on $R / J$ . Also $\sqrt{J} = m$ means some power $m^{N}$ lies in $J$ , hence kills $R / J$ . Therefore [L2] makes $J$ a $m$ -primary ideal.

L1L2given

1.2

Conversely, if $J$ is $m$ -primary, [L2] applied to the finite module $R / J$ gives $\sqrt{J} = m$ . Then [L1] says $(x_{1}, \dots, x_{d})$ is a system of parameters.

L1L2given

2.1

Thus parameter ideals are exactly the $d$ -generated $m$ -primary ideals.

step 1.1step 1.2∎

Depends on

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Sources

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., §21 (standard reference, not scraped)
The Stacks Project, Section 10.60: Dimension (standard reference, not scraped)
Melvin Hochster, Dimension theory and systems of parameters (standard reference, not scraped)