Volkov, M. V. 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 Keywords:variety of associative rings; identity; lattice of subvarieties; finite critical rings PDFBibTeX XMLCite \textit{M. V. Volkov}, Sib. Math. J. 25, 849--855 (1984; Zbl 0584.16010); translation from Sib. Mat. Zh. 25, No. 6(148), 23--30 (1984) Full Text: DOI 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 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.