Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial
Why this work is in the frame
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Bibliographic record
Abstract
Abstract If C = C ( R ) denotes the center of a ring R and g ( x ) is a polynomial in C [ x ], Camillo and Simón called a ring g ( x )-clean if every element is the sum of a unit and a root of g ( x ). If V is a vector space of countable dimension over a division ring D , they showed that end D V is g ( x )-clean provided that g ( x ) has two roots in C ( D ). If g ( x ) = x – x 2 this shows that end D V is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that end R M is g ( x )-clean for any semisimple module M over an arbitrary ring R provided that g ( x ) ∈ ( x – a )( x – b ) C [ x ] where a , b ∈ C and both b and b – a are units in R .
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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 | 0.001 |
| 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) | 0.002 | 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 it