On positive and critical theories of some classes of rings. (English. Russian original) Zbl 0951.17001

Sib. Math. J. 41, No. 2, 224-228 (2000); translation from Sib. Mat. Zh. 41, No. 2, 278-283 (2000).
Let \(\mathfrak X\) be some variety of rings defined by polyhomogeneous identities. The author proves that the positive theories of the following varieties coincide: (i) \(\mathfrak X\); (ii) the free ring in \(\mathfrak X\) of countable rank \(F\mathfrak X\); and (iii) the class \(\mathfrak X\cap F\) of all finite rings in \(\mathfrak X\).
Let \(\mathfrak N\) be an arbitrary variety of semigroups, groups, rings, or algebras over a field. Let \(F\mathfrak N\) be a free object in the variety \(\mathfrak N\) of countable rank and let \(\mathfrak N\cap F\) be the class of all free objects of the variety. The authors show that the positive theory of the algebraic system \(F\mathfrak N\) and that of the class \(\mathfrak N\cap F\) coincide.
As a corollary, the authors give a negative answer to Bokut’s question on decidability of the word problem for the class of finite Jordan rings.


17A01 General theory of nonassociative rings and algebras
17A30 Nonassociative algebras satisfying other identities
Full Text: DOI EuDML


[1] Vazhenin Yu. M. andRozenblat B. V., ”Decidability of the positive theory of free countably generated semigroups,” Mat. Sb.,116, No. 1, 120–127 (1981). · Zbl 0472.20020
[2] Rozenblat B. V., ”Permutational varieties of semigroups,” in: Abstracts: XVII All-Union Algebraic Conference. Part 2, Minsk, 1983, Belarus. Univ., Minsk, 1983, pp. 195–196.
[3] Bokut’ L. A., ”Algorithmic problems and embedding theorems: some open questions for rings, groups, and semi-groups,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 11, 3–11 (1982). · Zbl 0513.17001
[4] Makanin G. S., ”The decidability problem for equations in a free semigroup,” Mat. Sb.,103, No. 2, 147–237 (1977). · Zbl 0371.20047
[5] Roman’kov V. A., ”Undecidability of the endomorphic reducibility problem in free nilpotent groups and in free rings,” Algebra i Logika,16, No. 4, 457–471 (1977). · Zbl 0411.20021
[6] Vazhenin Yu. M., ”Critical theories for certain classes of nonassociative rings,” Algebra i Logika,28, No. 4, 393–401 (1989). · Zbl 0725.03004
[7] Vazhenin Yu. M. andPopov V. Yu., ”Critical theories of some types of free nilpotent rings,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 74–76 (1991). · Zbl 0737.03012
[8] Popov V. Yu., ”Decidable theories of Jordan rings,” in: Abstracts: X All-Union Conference on Mathematical Logic, Alma-Ata, 1990, p. 134.
[9] Popov V. Yu., ”Decidable theories of alternative rings,” in: Proceedings of the Soviet-French Colloquium on the Theory of Models, Karaganda, 1990, p. 41.
[10] Vazhenin Yu. M., ”Critical theories,” Sibirsk. Mat. Zh.,29, No. 1, 23–31 (1988). · Zbl 0638.03013
[11] Popov V. Yu., ”Critical theories of supervarieties of the variety of commutative associative rings,” Sibirsk. Mat. Zh.,36, No. 6, 1364–1374 (1995). · Zbl 0857.17004
[12] Bakhturin Yu. A. andOl’shanski├« A. Yu., Algebra-2. Identities. Contemporary Problems of Mathematics. Vol. 18 (Itogi Nauki i Tekhniki) [in Russian], VINITI, Moscow (1988). · Zbl 0664.17008
[13] Makanin G. S., ”Equations in a free group,” Izv. Akad. Nauk SSSR Ser. Mat.,46, No. 6, 1199–1273 (1982). · Zbl 0511.20019
[14] Zamyatin A. P. and Rozenblat B. V., ”Elementary theories of relatively free semigroups with a permutational identity,” in: Study on Algebraic Systems by Using the Properties of Their Subsystems [in Russian], Sverdlovsk, 1985, pp. 58–69. · Zbl 0633.20039
[15] Umirbaev U. U., ”The word problem for Jordan and right-alternative, algebras,” in: Some Questions and Problems of Analysis and Algebra, Novosibirsk Univ., Novosibirsk, 1985, pp. 120–127.
[16] Vazhenin Yu. M., Sets, Logic, and Algorithms [in Russian], Ural’sk. Univ., Ekaterinburg (1997).
[17] Keisler H. J. andChang C. C., Model Theory [Russian translation], Mir, Moscow (1977).
[18] Keisler H. J., ”Limit ultraproducts,” J. Symbolic Logic,30, 212–234 (1965). · Zbl 0147.25603
[19] Lyndon R. C., ”Properties preserved under homomorphism,” Pacific J. Math.,9, 143–154 (1959). · Zbl 0093.01101
[20] Malcev A. I., Algebraic Systems [in Russian], Nauka, Moscow (1970). · JFM 62.1103.02
[21] Lyndon R. C., ”Properties preserved in subdirect products,” Pacific. J. Math.,9, 155 (1959). · Zbl 0093.01102
[22] Los J., ”On the extending of models. I,” Fund. Math.,42, 38–54 (1955). · Zbl 0065.00401
[23] Tarski A., ”Contributions to the theory of models. I, II,” Koninkl. Nederl. Akad. Wetensch. Proc. Ser. A,57, 572–588 (1954). · Zbl 0058.24702
[24] Kharlampovich O. G. andSapir M. V., ”Algorithmic problems in varieties,” Internat. J. Algebra Comput.,5, No. 4–5, 379–602 (1995). · Zbl 0837.08002
[25] Zhevlakov K. A., Slin’ko A. M., Shestakov I. P., andShirshov A. I., Rings That Are Nearly Associative [in Russian], Nauka, Moscow (1978). · Zbl 0445.17001
[26] Shirshov A. I., ”On specialJ-rings,” Mat. Sb.,38, No. 2, 149–166 (1956). · Zbl 0070.02902
[27] Kharlampovich, O. G., ”A universal theory of some classes of Lie rings,” in: Study on Algebraic Systems [in Russian], Ural’sk. Univ., Sverdlovsk, 1984, pp. 156–164.
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