Free algebras and free groups in Ore extensions and free group algebras in division rings
Bibliographic record
Abstract
Let K be a field of characteristic zero, let σ be an automorphism of K and let δ be a σ-derivation of K. We show that the division ring D=K(x;σ,δ) either has the property that every finitely generated subring satisfies a polynomial identity or D contains a free algebra on two generators over its center. In the case when K is finitely generated over a subfield k we then see that for σ a k-algebra automorphism of K and δ a k-linear derivation of K, K(x;σ) having a free subalgebra on two generators is equivalent to σ having infinite order, and K(x;δ) having a free subalgebra is equivalent to δ being nonzero. As an application, we show that if D is a division ring with center k of characteristic zero and D⁎ contains a solvable subgroup that is not locally abelian-by-finite, then D contains a free k-algebra on two generators. Moreover, if we assume that k is uncountable, without any restrictions on the characteristic of k, then D contains the k-group algebra of the free group of rank two.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".