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
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".