Rings in which Every Element is a Sum of Two Tripotents
Why this work is in the frame
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Bibliographic record
Abstract
Abstract Let R be a ring. The following results are proved. (1) Every element of R is a sum of an idempotent and a tripotent that commute if and only if R has the identity x 6 = x 4 if and only if R ≅ R 1 × R 2 , where R 1 / J ( 2 1 ) is Boolean with U ( R 1 ) a group of exponent 2 and R 2 is zero or a subdirect product of ℤ 3 ’s. (2) Every element of R is either a sum or a difference of two commuting idempotents if and only if R ≅ R 1 × R 2 , where R 1 / J ( R 1 ) is Boolean with J ( R 1 ) = 0 or J ( R 1 ) = {0, 2} and R 2 is zero or a subdirect product of ℤ 3 ’s. (3) Every element of R is a sum of two commuting tripotents if and only if R ≅ R 1 × R 2 × R 3 , where R 1 / J ( R 1 ) is Boolean with U ( R 1 ) a group of exponent 2, R 2 is zero or a subdirect product of ℤ 3 ’s, and R 3 is zero or a subdirect product of ℤ 5 ’s.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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) | 0.013 | 0.003 |
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