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Distributivity of certain lattices of varieties of associative rings. (English. Russian original) Zbl 0584.16010

Sib. Math. J. 25, 849-855 (1984); translation from Sib. Mat. Zh. 25, No. 6(148), 23-30 (1984).
Let \({\mathfrak M}\) be a variety of associative rings which satisfies the identity \(x^ 2-x^ 3f(x)=0\), f(x)\(\in {\mathbb{Z}}[x]\). The author proves that the lattice of subvarieties of \({\mathfrak M}\) is distributive. The proof uses the structure of finite critical rings.
Reviewer: Yu.N.Mal’tsev

MSC:

16Rxx Rings with polynomial identity
08B15 Lattices of varieties
06D05 Structure and representation theory of distributive lattices
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References:

[1] A. Z. Anan’in and A. R. Kemer, ?Varieties of associative rings whose lattices of subvarieties are distributive,? Sib. Mat. Zh.,17, No. 4, 723-730 (1976).
[2] Dnestrovskaya Textbook. Unsolved Problems in the Theory of Rings and Modules [in Russian], Novosibirsk (1976).
[3] M. V. Volkov, ?Varieties of associative rings with conditions on the lattice of subvarieties,? Manuscript deposited with VINITI, No. 3805-79.
[4] Yu. N. Mal’tsev, ?Examples of varieties of associative rings,? Algebra Logika,19, No. 6, 669-675 (1980). · Zbl 0469.16011
[5] Yu. N. Mal’tsev and I. V. L’vov, ?Lattice of var M2 (GF(pn))? Fifteenth All-Union Algebraic Conference. Reports of Papers [in Russian], Krasnoyarsk (1979), p. 95.
[6] I. V. L’vov, ?On varieties of associative rings. I,? Algebra Logika,12, No. 3, 269-297 (1973).
[7] J. Cossey, ?Critical groups and the lattice of varieties,? Proc. Am. Math. Soc.,20, No. 1, 217-221 (1969). · Zbl 0186.03704 · doi:10.1090/S0002-9939-1969-0232824-0
[8] Yu. N. Mal’tsev, ?Varieties of associative algebras whose lattice of subvarieties is not distributive,? in: Algebraic Structures [in Russian], Kishinev (1980), pp. 110-112.
[9] B. M. Vernikov and M. V. Volkov, ?Almost chain varieties of alternative rings,? in: Studies in Contemporary Algebra [in Russian], Sverdlovsk (1979), pp. 22-39. · Zbl 0435.17010
[10] M. V. Volkov, ?Lattices of varieties of algebras,? Mat. Sb.,109, No. 1, 60-79 (1979). · Zbl 0411.17008
[11] Yu. N. Mal’tsev, ?Critical algebras,? Algebra Logika,20, No. 2, 155-164 (1981).
[12] Hanna Neumann, Varieties of Groups [Russian translation], Mir, Moscow (1969).
[13] G. Birkhoff, Lattice Theory [Russian translation], IL, Moscow (1952).
[14] M. V. Volkov, ?Periodic varieties of associative rings,? Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 3-13 (1979). · Zbl 0417.16005
[15] A. I. Mal’tsev, ?Multiplication of classes of algebraic systems,? Sib. Mat. Zh.,8, No. 2, 346-365 (1967).
[16] M. V. Volkov, ?The finiteness of the basis of identities for certain associative rings,? Sib. Mat. Zh.,22, No. 4, 79-87 (1981). · Zbl 0476.16017
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