Bibliographic record
Abstract
A ring [Formula: see text] is called left comorphic if, for each [Formula: see text] there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] Examples include (von Neumann) regular rings, and [Formula: see text] for a prime [Formula: see text] and [Formula: see text] One motivation for studying these rings is that the comorphic rings (left and right) are just the quasi-morphic rings, where [Formula: see text] is left quasi-morphic if, for each [Formula: see text] there exist [Formula: see text] and [Formula: see text] in [Formula: see text] such that [Formula: see text] and [Formula: see text] If [Formula: see text] here the ring is called left morphic. It is shown that [Formula: see text] is left comorphic if and only if, for any finitely generated left ideal [Formula: see text] there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] Using this, we characterize when a left comorphic ring has various properties, and show that if [Formula: see text] is local with nilpotent radical, then [Formula: see text] is left comorphic if and only if it is right comorphic. We also show that a semiprime left comorphic ring [Formula: see text] is semisimple if either [Formula: see text] is left perfect or [Formula: see text] has the ACC on [Formula: see text] After a preliminary study of left comorphic rings with the ACC on [Formula: see text] we show that a quasi-Frobenius ring is left comorphic if and only if every right ideal is principal; if and only if every left ideal is a left principal annihilator. We characterize these rings as follows: The following are equivalent for a ring [Formula: see text] [Formula: see text] is quasi-Frobenius and left comorphic. [Formula: see text] is left comorphic, left perfect and right Kasch. [Formula: see text] is left comorphic, right Kasch, with the ACC on [Formula: see text] [Formula: see text] is left comorphic, left mininjective, with the ACC on [Formula: see text] Some examples of these rings are given.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.013 | 0.002 |
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".