Bibliographic record
Abstract
A ring is called clean (resp., uniquely clean) if each of its elements can be (resp., uniquely) expressed as the sum of an idempotent and a unit. Motivated by recent work on uniquely clean rings in [6 Nicholson , W. K. , Zhou , Y. ( 2004 ). Rings in which elements are uniquely the sum of an idempotent and a unit . Glasg. Math. J. 46 : 227 – 236 .[Crossref], [Web of Science ®] , [Google Scholar]], we introduce the clean index of a ring R. For a ∈ R, let ℰ(a) = {e ∈ R: e 2 = e, a − e ∈ U(R)} where U(R) is the group of units of R and the clean index of R, denoted in(R), is defined by in(R) = sup{|ℰ(a)|: a ∈ R}. Thus, R is uniquely clean if and only if R is clean with in(R) = 1. So far, uniquely clean rings are the only clean rings whose structure is fully understood (see [6 Nicholson , W. K. , Zhou , Y. ( 2004 ). Rings in which elements are uniquely the sum of an idempotent and a unit . Glasg. Math. J. 46 : 227 – 236 .[Crossref], [Web of Science ®] , [Google Scholar]]). In this article, we characterize the (arbitrary) rings of clean indices 1, 2, 3 and determine the abelian rings of finite clean index. Applications to semipotent rings, semiprime rings, and clean rings are discussed.
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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 | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.005 | 0.003 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.005 | 0.011 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 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".