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

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Every finite-dimensional Noetherian local ring has a system of parameters

Statement

Let $(R, m)$ be a finite-dimensional Noetherian local ring, with $d = \dim R$ . Then $R$ has a system of parameters.

Facts & Assumptions

Given: A finite-dimensional Noetherian local ring $(R, m)$ of dimension $d$ .

[L1]

A first parameter can be chosen in $m$ outside every minimal prime, and for such a choice the quotient has dimension at most $d - 1$ (Choose a parameter that misses the top-dimensional minimal components).

[L2]

A system of parameters is a $d$ -tuple whose generated ideal has radical $m$ (Systems of parameters and parameter ideals).

[L3]

Prime ideals of a quotient correspond to primes upstairs containing the quotient ideal (Prime ideals of a quotient ring are exactly the prime ideals containing the ideal).

Proof

technique · induction on the dimension

1.1

If $d = 0$ , the empty tuple is a system of parameters by [L2].

L2basegiven

1.2

Assume $d > 0$ and that the theorem is known in dimensions $< d$ . By [L1], choose $x \in m$ outside every minimal prime. Then $R / (x)$ is a Noetherian local ring of dimension at most $d - 1$ . By the induction hypothesis, choose a system of parameters $({\overset{ˉ}{x}}_{2}, \dots, {\overset{ˉ}{x}}_{d})$ in $R / (x)$ of length $d - 1$ . Lift those elements to $x_{2}, \dots, x_{d} \in m$ .

L1L3ihchoose

2.1

The ideal generated by $({\overset{ˉ}{x}}_{2}, \dots, {\overset{ˉ}{x}}_{d})$ has radical $m / (x)$ in $R / (x)$ , so [L3] says the ideal $(x, x_{2}, \dots, x_{d})$ has radical $m$ in $R$ . By [L2], the tuple $(x, x_{2}, \dots, x_{d})$ is a system of parameters.

L2L3step 1.2

3.1

Therefore every finite-dimensional Noetherian local ring has a system of parameters.

step 1.1step 2.1discharge-induction∎

Depends on

Used by

Local dimension is the minimal number of generators of an ideal with maximal radical Theorem

Dependency tree · two levels

11 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

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., §21 (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, v4.03, §21 (standard reference, not scraped)
Melvin Hochster, Dimension theory and systems of parameters (standard reference, not scraped)