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Regular separably closed rings. (Italian. English summary) Zbl 0598.13009

It is well known that Pierce’s theory of Boolean localization [R. S. Pierce, Mem. Am. Math. Soc. 70 (1967; Zbl 0152.026)] is useful in various generalizations of the properties of fields for the case of commutative von Neumann regular rings. The author gives the following interesting example of such a generalization. As it was observed by Yu. L. Ershov [Sov. Math., Dokl. 8, 575-576 (1967); translation from Dokl. Akad. Nauk SSSR 174, 19-20 (1967; Zbl 0153.372)], the theories of separable closed fields with certain additional structure are model complete, and replacing the \(\{\) 0,1\(\}\)-valued functional symbols (from this structure) by idempotent-valued (i.e. Boolean-valued) functional symbols, the author obtains model complete theories of ”separably closed regular rings”. As in the case of fields this may be considered as a classification of such theories. The author considers also the connections between separably closed regular rings and differentially closed regular rings.
Reviewer: G.Dzhanelidze

MSC:

13L05 Applications of logic to commutative algebra
13B05 Galois theory and commutative ring extensions
03C60 Model-theoretic algebra
12F10 Separable extensions, Galois theory
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
12L99 Connections between field theory and logic
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References:

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