Free algebras and free groups in Ore extensions and free group algebras in division rings
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it