On the sheaf characterization of Gelfand rings
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Bibliographic record
Abstract
This paper deals with the commutative rings with unit in which a+b = 1 implies that (1+ar)(1+bs) = 0 for suitable r and s. These rings have been considered, alternatively, in terms of conditions on their maximal ideals (DeMarco-Orsatti 1971, Mulvey 1979, Johnstone 1982) which are equivalent to the above in the presence of the Axiom of Choice (Banaschewski 2000). The purpose here is to give new characterizations of these rings as the rings of global elements of certain sheaves of rings on compact regular frames, the pointfree counterparts of compact Hausdorff spaces, augmenting a previous result of this type (Banaschewski 2000). In particular, one of the present cases provides the exact pointfree version of (the commutative case of) the original characerization of these rings of Mulvey (1979) which the earlier work did not achieve.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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