Rings in which Every Element is a Sum of Two Tripotents
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 x6 = x4 if and only if R ≅ R1 × R2, where R1/J(21) is Boolean with U(R1) a group of exponent 2 and R2 is zero or a subdirect product of ℤ3’s. (2) Every element of R is either a sum or a difference of two commuting idempotents if and only if R ≅ R1 × R2, where R1/J(R1) is Boolean with J(R1) = 0 or J(R1) = {0, 2} and R2 is zero or a subdirect product of ℤ3’s. (3) Every element of R is a sum of two commuting tripotents if and only if R ≅ R1 × R2 × R3, where R1/J(R1) is Boolean with U(R1) a group of exponent 2, R2 is zero or a subdirect product of ℤ3’s, and R3 is zero or a subdirect product of ℤ5’s.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".