Köthe’s matrix conjecture for nil algebras
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Köthe’s matrix question asks whether matrices over nil algebras must be nilpotent. The counterexample target is a nil algebra over the algebraic closure of F₂ containing a nonnilpotent 2×2 matrix; after unitization the nil algebra is a two-sided ideal.
Why it matters and what counts as progress
Why it matters: Through Krempa’s classical implication, a matrix-form counterexample would also obstruct Köthe’s original conjecture.
What counts as progress: Construct a matrix over a nil algebra that is not nilpotent, or prove the matrix implication under a restricted ring hypothesis.
How the problem developed
- 1930 · Conjecture [1]
Gottfried Köthe Asked whether the sum of two nil left ideals in a ring must be nil.
- 1972 · Equivalent formulation [1]
Gerhard Krempa Showed the conjecture equivalent to the assertion that matrix ideals over nil two-sided ideals remain nil.
GPT-6 Astra’s autonomous construction, reported by Tom Adamczewski, Bernhard Böhmler and René Marczinzik Released a counterexample to Köthe’s conjecture through a non-nilpotent matrix in a matrix ideal over a nil algebra; the paper generalizes the construction to countable fields.
AI contribution
AI-found counterexample to Köthe’s matrix conjecture
GPT-6 Astra constructed a nil algebra over the algebraic closure of F₂ with a nonnilpotent 2×2 matrix. In its unitization, the algebra is a nil two-sided ideal, refuting Krempa’s matrix form of Köthe’s conjecture.
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