ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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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The module $R / (x^{i})$ over $k [x] / (x^{n})$ has length $i$

Example

Let $R = k [x] / (x^{n})$ with $n \geq 1$ , and let $1 \leq i \leq n$ . Then the $R$ -module $R / (x^{i})$ has length $i$ .

Facts & Assumptions

Given: A field $k$ , integers $n \geq 1$ and $1 \leq i \leq n$ , and the ring $R = k [x] / (x^{n})$ .

Verification

technique · direct

1.1

By The truncated polynomial ring $k [x] / (x^{n})$ is local Artinian of length $n$ , every quotient $(x^{m}) / (x^{m + 1})$ with $0 \leq m < n$ is one-dimensional over $k$ . In particular each such quotient has length $1$ , and the module $R / (x) ≅ k$ also has length $1$ .

givenalgebra

2.1

For every $1 \leq m < i$ , the natural projection $R / (x^{m + 1}) \to R / (x^{m})$ has kernel $(x^{m}) / (x^{m + 1})$ , so there is a short exact sequence $0 \to (x^{m}) / (x^{m + 1}) \to R / (x^{m + 1}) \to R / (x^{m}) \to 0.$ Starting from the base case $ℓ_{R} (R / (x)) = 1$ from step 1.1 and applying Module length is additive in short exact sequences inductively, one gets $ℓ_{R} (R / (x^{m + 1})) = ℓ_{R} (R / (x^{m})) + 1 = m + 1$ for every $m < i$ .

step 1.1giveninduction

3.1

Taking $m = i - 1$ in step 2.1 gives $ℓ_{R} (R / (x^{i})) = i$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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., Exercise (19.10) (standard reference, not scraped)
The Stacks Project, Section 10.52: Length (standard reference, not scraped)

The module R/(xi)R/(xi) over k[x]/(xn)k[x]/(xn) has length ii