Bibliographic record
Abstract
Friedman, Simpson, and Smith showed that, over RCA$_0$, the statements “Every ring has a maximal ideal” and “Every ring has a prime ideal” are equivalent to ACA$_0$ and WKL$_0$, respectively. More recently, Downey, Lempp, and Mileti have shown that, over RCA$_0$, the statement “Every ring that is not a field contains a nontrivial ideal” is equivalent to WKL$_0$. In this article we explore the reverse mathematical strength of the classic theorems from commutative algebra which say that every Artinian ring is Noetherian, and every Artinian ring is of finite length. In particular we show that, over RCA$_0$, the former implies WKL$_0$ and is implied by ACA$_0$, while over RCA$_0$+B$\Sigma _2$, the latter is equivalent to ACA$_0$.
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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 | 0.003 | 0.010 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.004 | 0.012 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.013 | 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".