DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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Systems of parameters and parameter ideals

Definition

Let $(R, m)$ be a finite-dimensional Noetherian local ring of dimension $d = \dim R < \infty$ . A tuple

(x_{1}, \dots, x_{d}) \in m^{d}

is a system of parameters when

\sqrt{(x_{1}, \dots, x_{d})} = m .

Equivalently, the ideal generated by the tuple has maximal radical. That ideal is called a parameter ideal.

When $d = 0$ , the empty tuple is a system of parameters exactly when $\sqrt{(0)} = m$ . In a zero-dimensional Noetherian local ring this is the correct convention, so systems of parameters still have length equal to the dimension.

Depends on

Used by

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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)