Bibliographic record
Abstract
A ring is called left quasi-duo (left QD) if every maximal left ideal is a right ideal, and it is called I-finite if it contains no infinite orthogonal set of idempotents. It is shown that a ring is I-finite and left QD if and only if it is a generalized upper-triangular matrix ring with all diagonal rings being division rings except the lower one, which is either a division ring or it is I-finite, left QD and left `soclin' (left QDS). Here a ring is called left soclin if each simple left ideal is nilpotent. The left QDS rings are shown to be finite direct products of indecomposable left QDS rings, in each of which 1=f1+⋯+fm1=f1+⋯+fm where the fifi are orthogonal primitive idempotents, with fk≈flfk≈fl for all k,l,k,l, and ≈≈ is the block equivalence on {f1,…,fm}.{f1,…,fm}.A ring is shown to be left soclin if and only if every maximal left ideal is left essential, if and only if the left socle is contained in the left singular ideal. These left soclin rings are proved to be a Morita invariant class; and if a ring is semilocal and non-semisimple, then it is left soclin if and only if the Jacobson radical is essential as a left ideal..Left quasi-duo elements are defined for any ring and shown to constitute a subring containing the centre and the Jacobson radical of the ring. The `width' of any left QD ring is defined and applied to characterize the semilocal left QD rings, and to clarify the semiperfect case..
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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 | 0.001 |
| 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 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".