False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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.

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

False statement: every semisimple ring is commutative

Statement

False claim: every semisimple ring is commutative. See $M_{n} (F)$ as a direct sum of minimal left ideals.

Facts & Assumptions

Given: The hypotheses and objects in the false claim.

[L1]

For every field $F$ and $n \geq 1$ , the left regular module of $M_{n} (F)$ is the internal direct sum $M_{n} (F) = ⨁_{j = 1}^{n} M_{n} (F) e_{j j}$ of simple left ideals. ( $M_{n} (F)$ as a direct sum of minimal left ideals).

[L2]

For every field $F$ and every natural $n \geq 2$ , the ring $M_{n} (F)$ is not commutative. ( $M_{n} (F)$ is noncommutative for every $n \geq 2$ ).

Refutation

technique · direct

1.1

For any field $F$ , the preceding decomposition makes $M_{2} (F)$ semisimple, while two matrix units multiply in opposite orders to different results.

L1L2givenalgebra

2.1

The witness does not depend on the characteristic: in $M_{2} (F)$ one has $e_{11} e_{12} = e_{12}$ and $e_{12} e_{11} = 0$ , and $e_{12} \neq 0$ in every field, characteristic two included, so the two products differ there as well. This proves the stated claim.

step 1.1L2givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 6 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

William Crawley-Boevey, Noncommutative Algebra, Chapter 1 Sections 1.1-1.9 (standard reference, not scraped)