Nil clean group rings over metacyclic groups
Bibliographic record
Abstract
A ring is nil clean if each of its elements is the sum of an idempotent and a nilpotent. Previously in [J. Cui, Y. Li and H. Wang, On nil clean group rings, Commun. Algebra 49(2) (2021) 790–796], the nil clean group rings over dihedral groups and generalized quaternion groups were completely determined. In this paper, we investigate group rings [Formula: see text] over more general metacyclic groups [Formula: see text], where [Formula: see text] We provide a new criterion for a general nil clean group ring based on the upper central series of [Formula: see text] consisting of [Formula: see text]-groups, and whether the group ring over its respective factor group is nil clean. We also determine values of [Formula: see text] and [Formula: see text] (as a prime power) in order for [Formula: see text] to be nil clean, and verify whether their corresponding group rings are indeed nil clean.
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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".