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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.

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.

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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 2026-09-28}
}

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Claims
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Disputed
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Refuted
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On the literature board
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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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.