TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

DCC and minimal-condition characterizations of Artinian modules

Statement

For a left $R$ -module $M$ , DCC is equivalent to the condition that every nonempty family of submodules has a minimal member. The implication from DCC to the minimal condition uses dependent choice. See Artinian modules by the descending chain condition.

Facts & Assumptions

Given: The hypotheses and objects in the Statement. The adopted axiom of dependent choice is assumed for the one direction identified in the Statement; it is not cited as a forward dependency.

[L1]

A left $R$ -module $M$ is Artinian when every descending chain $M_{0} \supseteq M_{1} \supseteq \dots$ of submodules stabilizes: there is $N$ such that $M_{n} = M_{N}$ for all $n \geq N$ . This is the descending chain condition. (Artinian modules by the descending chain condition).

Proof

technique · direct

1.1

DCC gives a minimal member of every nonempty family by contradiction: under dependent choice, absence of a minimal member yields a strict descending chain.

L1givenalgebra

2.1

Conversely, a nonstabilizing descending chain has no minimal member.

step 1.1givenalgebra

3.1

Repeated terms are what make step 2.1 correct rather than an equivocation: minimality of $M_{k}$ in the family ${M_{n}}$ means no member is properly contained in it, so $M_{n} = M_{k}$ for all $n \geq k$ , which is stabilization. For the zero module the only submodule is $0$ , every nonempty family is ${0}$ with minimal member $0$ , and every descending chain is constant, so both conditions hold. This proves the stated claim.

step 2.1givenalgebra∎

Depends on

Artinian modules by the descending chain condition

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 4 results over 3 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Arvind Nair, Algebra I, Lecture 5 (standard reference, not scraped)