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 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) | 0.000 | 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".