On clean, weakly clean and feebly clean commutative group rings
Bibliographic record
Abstract
A ring [Formula: see text] is said to be clean if each element of [Formula: see text] can be written as the sum of a unit and an idempotent. [Formula: see text] is said to be weakly clean if each element of [Formula: see text] is either a sum or a difference of a unit and an idempotent, and [Formula: see text] is said to be feebly clean if every element [Formula: see text] can be written as [Formula: see text], where [Formula: see text] is a unit and [Formula: see text] are orthogonal idempotents. Clearly, clean rings are weakly clean rings and both of them are feebly clean. In a recent paper (J. Algebra Appl. 17 (2018) 1850111 (5 pp.)), McGoven characterized when the group ring [Formula: see text] is weakly clean and feebly clean, where [Formula: see text] are distinct primes. In this paper, we consider a more general setting. Let [Formula: see text] be an algebraic number field, [Formula: see text] its ring of integers, [Formula: see text] a nonzero prime ideal and [Formula: see text] the localization of [Formula: see text] at [Formula: see text]. We investigate when the group ring [Formula: see text] is weakly clean and feebly clean, where [Formula: see text] is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when [Formula: see text] is a cyclotomic field or [Formula: see text] is a quadratic field.
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.003 | 0.007 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 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".