Toffalori, Carlo Regular separably closed rings. (Italian. English summary) Zbl 0598.13009 Rend. Sem. Mat. Univ. Padova 71, 15-33 (1984). 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 Keywords:von Neumann regular rings; model complete; separably closed regular rings; differentially closed regular rings Citations:Zbl 0152.026; Zbl 0153.372 PDFBibTeX XMLCite \textit{C. Toffalori}, Rend. Semin. Mat. Univ. Padova 71, 15--33 (1984; Zbl 0598.13009) Full Text: Numdam EuDML References: [1] G. Cherlin , Model-theoretic algebra . Selected topics, Lecture Notes in Mathematics 521 , Springer , 1975 . MR 539999 | Zbl 0332.02056 · Zbl 0332.02056 · doi:10.1007/BFb0079565 [2] G. Cherlin - J. Reineke , Categoricity and stability of commutative rings , Ann. Math. Logic , 10 ( 1976 ), pp. 367 - 399 . MR 480007 | Zbl 0326.02041 · Zbl 0326.02041 · doi:10.1016/0003-4843(76)90017-6 [3] Y. Ersov , Fields with a solvable theory , Dokl. Akad. Nauk SSSR , 174 ( 1967 ), pp. 19 - 20 ; traduzione inglese in Sov. Math. , 8 ( 1967 ), pp. 575 - 576 . MR 214575 | Zbl 0153.37201 · Zbl 0153.37201 [4] N. Jacobson , Lectures in abstract algebra , vol. III , Van Nostrand , 1964 . MR 172871 · Zbl 0124.27002 [5] S. Lang , Introduction to algebraic geometry , Addison-Wesley , 1958 . MR 100591 | Zbl 0247.14001 · Zbl 0247.14001 [6] R.S. Pierce , Modules over commutative regular rings , Mem. Amer. Math. Soc. , 70 , 1967 . MR 217056 | Zbl 0152.02601 · Zbl 0152.02601 [7] D. Ponasse - J.C. Carrega , Algébre et topologie booléennes , Masson , 1979 . MR 532013 | Zbl 0465.06001 · Zbl 0465.06001 [8] C. Toffalori , Strutture differenzialmente chiuse per alcune classi di anelli , in corso di stampa. · Zbl 0485.03013 [9] V. Weispfenning , Model-completeness and elimination of quantifiers for subdirect products of structures , J. Algebra , 36 ( 1975 ), pp. 252 - 277 . MR 437339 | Zbl 0318.02052 · Zbl 0318.02052 · doi:10.1016/0021-8693(75)90101-5 [10] C. Wood , Notes on the stability of separably closed fields , J. Symbolic Logic , 44 ( 1979 ), pp. 412 - 416 . MR 540671 | Zbl 0424.03014 · Zbl 0424.03014 · doi:10.2307/2273133 [11] C. Wood , The model theory of differential fields revisited , Israel J. Math. , 25 ( 1976 ), pp. 331 - 352 . MR 457199 | Zbl 0346.02030 · Zbl 0346.02030 · doi:10.1007/BF02757008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.