False statementConstruction: AI-adaptedVerification: 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.

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

1.1L1L2givenalgebra

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

2.1step 1.1L2givenalgebra∎

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.

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

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