{"id":"openai-math/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic","pdf":"https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026","tags":["group-rings","direct-finiteness","algebra"],"score":0,"title":"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic","domain":"Algebra","family":"197","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.RA","summary":"We construct a counterexample to Kaplansky's direct-finiteness conjecture in odd characteristic. For one specified odd prime p , we obtain a field K of order p 4 , a finitely generated group G containing torsion, and finite sums $`a,b\\in K[G]`$ with $`ab=1`$ but $`ba\\ne1`$. The same elements define a cellular automaton on K G that is injective but not surjective.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-26","reactions":{},"formalized":true,"family_title":"A torsion-free group algebra that is not directly finite"}