TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

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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

1.1L1givenalgebra

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

2.1step 1.1givenalgebra

Conversely, a nonstabilizing descending chain has no minimal member.

3.1step 2.1givenalgebra∎

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.

Depends on

Artinian modules by the descending chain condition

Used by

Dependency tree · two levels

2 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

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