Existence of finitely presented nil but nonnilpotent algebras
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Can an infinite-dimensional nil associative algebra be finitely presented? The specific target is a nonunital algebra over F₂ that is Jacobson radical but not nilpotent, whose unitization remains finitely presented, algebraic and infinite-dimensional.
Why it matters and what counts as progress
Why it matters: Finite presentation imposes a stronger algebraic finiteness condition than the known existence of infinite-dimensional nil algebras.
What counts as progress: Construct such an algebra with all stated properties or prove an obstruction for a restricted presentation class.
How the problem developed
- 19 Sept 2008 · Open question [1]
Agata Smoktunowicz Recorded Ufnarovskij’s question whether every finitely presented nil algebra must be nilpotent, in a discussion of related nil-algebra problems.
- 23 Sept 2026 · Reported result [2]
OpenAI Reported a finitely presented nonunital nil associative algebra over F₂ that is Jacobson radical but not nilpotent; its unitization is finitely presented and algebraic.
AI contribution
An infinite finitely presented periodic group
The September OpenAI manuscript claims an infinite-dimensional finitely presented nonunital nil associative algebra over F₂ that is Jacobson radical but not nilpotent. Its unitization is finitely presented, algebraic and infinite-dimensional.
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Publication dates and editorial changes.
OpenAI reports a finitely presented nil algebra that is not nilpotent