Fields and rings with few types
Abstract
Let R be an associative ring with possible extra structure. R is said to be weakly small if there are countably many 1-types over any finite subset of R. It is locally P if the algebraic closure of any finite subset of R has property P. It is shown here that a field extension of finite degree of a weakly small field either is a finite field or has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. Every weakly small division ring of positive characteristic is locally finite dimensional over its centre. The Jacobson radical of a weakly small ring is locally nilpotent. Every weakly small division ring is locally, modulo its Jacobson radical, isomorphic to a product of finitely many matrix rings over division rings.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2011
- DOI:
- 10.48550/arXiv.1104.0288
- arXiv:
- arXiv:1104.0288
- Bibcode:
- 2011arXiv1104.0288M
- Keywords:
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- Mathematics - Logic
- E-Print:
- Journal of Symbolic Logic, Volume 78, Issue 1 (2013), 72-84