Bibliographic record
Abstract
A theorem of Faith and Walker asserts that a ring R is quasi-Frobenius if and only if every injective right R -module is projective and hence that every right module over a quasi-Frobenius ring embeds in a free module. There is an open problem here. If we call a ring R a right FGF ring if every finitely generated right R -module can be embedded in a free right R -module, it is not known if the following assertion is true: The FGF-Conjecture . Every right FGF ring is quasi-Frobenius Here are four important results on the conjecture: Every left Kasch, right FGF ring is quasi-Frobenius. Every right self-injective, right FGF ring is quasi-Frobenius. Every right perfect, right FGF ring is quasi-Frobenius. Every right CS, right FGF ring is quasi-Frobenius. We prove all these assertions; in fact we capture all of (1), (2), and (3) in Theorem 7.19: If M n (R) is a right C2 ring for each n ≥ 1 and every 2-generated right R -module embeds in a free module then R is quasi-Frobenius. This theorem also implies that the FGF-conjecture is true for right FP-injective rings, and it reformulates the conjecture by showing that it suffices to prove that every right FGF ring is a right C2 ring. Furthermore, the theorem shows that the conjecture is true for semiregular rings with Z r = J . We call these rings right weakly continuous, and investigate their basic properties.
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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 | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.022 | 0.006 |
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".