On Uniform Diagonalisation of Matrices over Regular Rings and One-Accessible Regular Algebras
Bibliographic record
Abstract
In connection with the fundamental Separativity Problem for regular rings, we show that a regular algebra R over a commutative ring admits a uniform diagonalisation formula where the entries of P and Q are algebra expressions in the a i and the a i ', if and only if R is strongly regular (abelian regular in the terminology of Goodearl, K.R. (1979 Goodearl, K. R. 1979. Von Neumann Regular Rings, London: Pitman. 2nd Malabar, Fl: Krieger. 1991 [Google Scholar]). Von Neumann Regular Rings. London: Pitman. 2nd ed. Krieger, Malabar, CFI. 1991). Next, we study regular algebras R over a field F such that for any a ∈ R there exist b ∈ F[a] and b' ∈ R such that bb'b = b, b'bb' = b' and the subalgebra of R generated by a and b' is regular. Such algebras are called one-accessible. We show that a finite product of matrix rings over a field is one-accessible and that a regular algebra over an uncountable perfect field is one-accessible if and only if it is algebraic. Tangentially, we elucidate and characterize when a nilpotent element has all its powers regular (or unit-regular) in an arbitrary algebra R over a commutative ring Λ. This involves finite direct products of matrix rings over factor rings of Λ.
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 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 | 0.004 |
| 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 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".