ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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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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Computing the length of $R / (π^{n})$

Example

Let $V$ be a discrete valuation ring with uniformiser $π$ . Then $0 \subset (π^{n - 1}) / (π^{n}) \subset (π^{n - 2}) / (π^{n}) \subset \dots \subset (π) / (π^{n}) \subset V / (π^{n})$ is a composition series, so $ℓ_{V} (V / (π^{n})) = n$ .

Facts & Assumptions

Given: A discrete valuation ring $V$ , a uniformiser $π$ , and an integer $n \geq 1$ .

[F1]

A uniformiser generates the maximal ideal of a DVR (Uniformising parameters).

[L1]

In a DVR one has $ℓ_{V} (V / (π^{n})) = n$ (Length and valuation in a DVR).

Verification

technique · direct

1.1

Each successive quotient in the displayed filtration is generated by the class of $π^{i}$ , so multiplication by $π^{i}$ identifies it with $V / (π)$ . Thus every factor has length $1$ .

F1givenalgebra

2.1

Adding the $n$ successive factors recovers the quotient $V / (π^{n})$ , and [L1] confirms that the total length is exactly $n$ .

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

J. S. Milne, A Primer of Commutative Algebra, v4.03, Discrete valuation rings after Example 20.1 (standard reference, not scraped)

Computing the length of R/(πn)R/(πn)