Bibliographic record
Abstract
A surprising amount of information about right self-injective rings can be obtained by studying a much larger class of rings, the right mininjective rings. A ring is called right mininjective if every isomorphism between two simple right ideals is given by left multiplication. The basic general facts about these rings are derived in this chapter and this work serves as a basis for the study of two important subclasses: the P-injective rings in Chapter 5 and the simple injective rings in Chapter 6. After giving several examples (including a right mininjective ring that is not left mininjective), we show that mininjectivity is a Morita invariant and that “min” versions of the C2- and C3-conditions hold. (The “min” version of the C1-condition is studied in Chapter 4.) Surprisingly, under a mild commutativity condition, a ring R is right minininjective if and only if its right socle is square-free, and every factor ring of R is right mininjective if and only if R has a distributive lattice of right ideals. In general, it is shown that the right socle of any right mininjective ring R is contained in the left socle (in fact, if kR is a simple right ideal then Rk is also simple). This remarkable fact is used repeatedly throughout the book. If a right mininjective ring is semiregular and has essential right socle, we show that the right singular ideal equals the Jacobson radical, extending the situation for right self-injective rings.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 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 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".