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Level A · Machine-checkable Hard Algebra P-koethe

Köthe conjecture

The Köthe conjecture: In any ring, the sum of two nil left ideals is nil.

Status (as of 2026-10-31): Disproved (September 2026). A counterexample to the Köthe conjecture was constructed with AI assistance (Epoch AI with an OpenAI model) and formalised in Lean; Böhmler and Marczinzik posted a human exposition (arXiv 2609.07996). The Formal Conjectures source still lists the statement as open.

Current state (tracked automatically)

  • Formal Conjectures marks the Lean statement KotheConjecture as open.
All tracked facts and sources
FactValueSource
Formal Conjectures: status of `KotheConjecture`research openFormal Conjectures
Formal Conjectures: research statements in the file5 open, 8 solvedFormal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

Settled outside Cairn Commons — see the status above and the sources below. Research tasks for this problem have stopped; a literature claim that cites the solution is still welcome.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

Submit a claim Follow (RSS)
Cite 
@misc{cairn-koethe,
  title        = {Köthe conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/koethe}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

The Köthe conjecture: In any ring, the sum of two nil left ideals is nil.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Koethe.

theorem KotheConjecture (I J : Ideal R) (hI : IsNil I) (hJ : IsNil J) : IsNil (I + J)

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • KotherConjecture.variants.le_KotherRadical — The Köthe conjecture: every left nil radical is contained in the Köthe radical.
  • KotherConjecture.variants.general_matrix — The Köthe conjecture: for any nil ideal I of R, the matrix ideal M_n(I) is a nil ideal of the matrix ring M_n(R).
  • KotherConjecture.variants.two_by_two_matrix — The Köthe conjecture: for any nil ideal I of R, the matrix ideal M_2(I) is a nil ideal of the matrix ring M_2(R).
  • KotherConjecture.variants.matrixOver_KotherRadical — The Köthe conjecture: for any finite index type n, the Köthe radical of the matrix ring M_n(R) is the ideal M_n(Nil*(R)) of matrices with entries in the Köthe…

References

Wikipedia

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.