Breakthrough Tracker record
A preprint claims a counterexample to Köthe's 90-year-old ring-theory conjecture
Tom Adamczewski, Bernhard Böhmler and René Marczinzik construct, over every countable field, a nil algebra whose 2-by-2 matrix ring is not nil. This yields two nil left ideals with a nonnil sum and therefore a counterexample to Köthe's conjecture, while also answering a question of Rowen negatively.
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- Stable ID
math-kothe-conjecture-counterexample-2026- Revision
math-kothe-conjecture-counterexample-2026.v1- Field
- Mathematics · Noncommutative ring theory and nil algebras
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional counterexample
- Last checked
AI role
No substantive AI role is disclosed in the inspected manuscript.
Record details
- Problem or result
- Köthe's conjecture that the sum of two nil left ideals is nil
- Authors
- Tom Adamczewski, Bernhard Böhmler and René Marczinzik
- Institutions
- Epoch AI; Leibniz University Hannover; University of Bonn
- Result date
- Preprint submitted September 7, 2026
Why it matters
Köthe's conjecture has resisted resolution since 1930 and is a foundational question about nil ideals and matrix rings in noncommutative algebra.
Limits
The construction is an unreviewed first-version preprint and applies over countable fields. No independent specialist verification or published version was located in this cycle.
Sources
Correction and revision history
- 2026-09-10 — Added after primary-source, scope, status, AI-role and limitation review.
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