Bibliographic record
Abstract
A type decomposition of a module M over a ring R is a direct sum decomposition for which any two distinct summands have no nonzero isomorphic submodules.In this paper, we investigate when a module possesses certain kinds of type decompositions and when such decompositions are unique.Introduction.It is well known that every torsion abelian group has a unique decomposition into its p-torsion subgroups.By Goodearl-Boyle [4], every nonsingular injective module E has a unique decom-are of types I, II, III respectively, see Definition 2.7.Why do such decompositions exist?Why are such decompositions unique?Are there any common things between these two results?All these questions will be answered in this paper.In fact, we can present a more general theory on existence and uniqueness of type decompositions of modules, so that the above results, as well as many other known results, are obtained as very special cases.The common property for certain diverse kinds of direct sum decompositions of modules M including the two decompositions above is that any two distinct direct summands have no nonzero isomorphic submodules, or equivalently all direct summands are what we will call type submodules.The cause for the existence of such decompositions is that these modules M have a 'decomposability property' which will be discussed in detail in Section 1, while the uniqueness of such direct sum decompositions is ensured by a module property called UTC.A theory of such modules is developed in Section 2. Throughout, all rings R are associative with identity and modules are unital right R-modules and M is an R-module.A class K of modules is a type, or natural class, if it is closed under isomorphic copies,
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".