Must a finitely presented nil algebra be nilpotent? (Ufnarovskij's question)
An associative algebra without identity is nil if every element has some power equal to zero, and nilpotent if for one . Golod's construction gives finitely generated nil algebras that are not nilpotent, and Lenagan, Smoktunowicz and Young built such algebras with restricted growth, but these are not finitely presented. Smoktunowicz records as Ufnarovskij's question whether a finitely presented nil algebra must be nilpotent; Amitsur asked the analogous question for finitely presented Jacobson radical algebras, and the finite-presentation version of Kurosh's algebraic finiteness problem asks the same for finitely presented algebraic algebras. Is every finitely presented nil algebra nilpotent?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Noncommutative ring theory; nil and radical algebras
- Posed by
- Ufnarovskij, as recorded by Smoktunowicz (Bull. LMS 2008, Question 8.1); also Lenagan, Smoktunowicz and Young (2012, Section 9); radical form due to Amitsur
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims Corollary 3.8: there is an infinite-dimensional finitely presented nonunital associative -algebra that is nil and Jacobson radical but not nilpotent; its unitization is finitely presented, algebraic and infinite-dimensional. This answers Ufnarovskij's nil question, Amitsur's radical question and the finite-presentation version of Kurosh's algebraic finiteness question negatively, over . Every matrix over its positive-degree ideal is nilpotent. It does NOT give examples over other fields or in characteristic zero, and nilpotence indices are unbounded.
What the AI did
The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. The nil algebra is the positive-degree ideal of the same finitely presented graded algebra used to build the periodic Steinberg group.
Verification
No independent mathematician has checked this yet. Corollary 3.8 and the introduction of the principal manuscript were read against the question: an infinite-dimensional finitely presented nonunital nil associative -algebra that is Jacobson radical but not nilpotent, whose unitization is finitely presented, algebraic and infinite-dimensional. This part has no Lean formalisation: lean/docs/247.md says the nil-algebra and radical-algebra conclusions are outside the formalised statements. The release README warns that some unformalised results could have issues.