A generalization of (D + M)-construction
Bibliographic record
Abstract
Let [Formula: see text] be a local domain with [Formula: see text] such that [Formula: see text]. Each subring [Formula: see text] of [Formula: see text] determines a subring [Formula: see text] of [Formula: see text]. This construction arises frequently in Algebra, essentially in connection with counter-examples. In this paper, we use the results of the authors [D. F. Anderson and A. Badawi, On [Formula: see text]-Prüfer rings and [Formula: see text]-Bezout rings, Houston J. Math. 30(2) (2004) 331–343; A. Badawi, On [Formula: see text]-chained rings and [Formula: see text]-pseudo-valuation rings, Houston J. Math. 27(4) (2001) 725–736; A. Badawi, On nonnil-Noetherian rings, Comm. Algebra 31(4) (2003) 1669–1677; A. Badawi, On pseudo-almost valuation domains, Comm. Algebra 35(4) (2007) 1167–1181; E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form [Formula: see text], Michigan Math. J. 20 (1973) 79–95; J. W. Brewer and E. A. Rutter, [Formula: see text] constructions with general overrings, Michigan Math. J. 23(1) (1976) 33–42; A. Esmaeelnezhad and P. Sahandi, On [Formula: see text]-pseudo almost valuation rings, Bull. Korean Math. Soc. 52(3) (2015) 935–946; R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers on Pure and Applied Mathematics, Vol. 90 (Queen’s University, Kingston, 1992); B. Khoualdia and A. Benhissi, Nonnil-coherent rings, Beitr. Algebra Geom. 57(2) (2016) 297–305; G. H. Tang, F. G. Wang and W. Zhao, On [Formula: see text]-Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] to generalize the above construction in the context of rings with prime Nilradical.
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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".