Noncommutative analogues of a cancellation theorem of Abhyankar, Eakin,\n and Heinzer
Bibliographic record
Abstract
Let $k$ be a field and let $A$ be a finitely generated $k$-algebra. The\nalgebra $A$ is said to be cancellative if whenever $B$ is another $k$-algebra\nwith the property that $A[x]\\cong B[x]$ then we necessarily have $A\\cong B$. An\nimportant result of Abhyankar, Eakin, and Heinzer shows that if $A$ is a\nfinitely generated commutative integral domain of Krull dimension one then it\nis cancellative. We consider the question of cancellation for finitely\ngenerated not-necessarily-commutative domains of Gelfand-Kirillov dimension\none, and show that such algebras are necessarily cancellative when the\ncharacteristic of the base field is zero. In particular, this recovers the\ncancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero\nwhen one restricts to the commutative case. We also provide examples that show\naffine domains of Gelfand-Kirillov dimension one need not be cancellative when\nthe base field has positive characteristic, giving a counterexample to a\nconjecture of Tang, the fourth-named author, and Zhang. In addition, we prove a\nskew analogue of the result of Abhyankar-Eakin-Heinzer, in which one works with\nskew polynomial extensions as opposed to ordinary polynomial rings.\n
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 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".