A note on β-classes
Bibliographic record
Abstract
A ring [Formula: see text] is an essential extension of its ideal [Formula: see text] if [Formula: see text], for every nonzero ideal [Formula: see text] of [Formula: see text]. Let [Formula: see text] be the prime radical. A class [Formula: see text] of prime rings that is closed under ideals and essential extensions is called a [Formula: see text]-class if [Formula: see text], where [Formula: see text] is the upper radical determined by [Formula: see text]. Let * be the class of all semiprime rings [Formula: see text] such that [Formula: see text] [Formula: see text] for every ideal [Formula: see text] of [Formula: see text] with [Formula: see text]. In this paper, we give some necessary and sufficient conditions for [Formula: see text] to be a [Formula: see text]-class. Intersections of [Formula: see text]-classes have always been found to be [Formula: see text]-classes and Leavitt asked whether this is always true. We show some of the strong restrictions which must apply to any possible counterexample.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.004 | 0.005 |
| Scholarly communication | 0.004 | 0.015 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.008 |
| Insufficient payload (model declined to judge) | 0.013 | 0.003 |
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".