A reduction theorem for ring varieties whose subvariety lattice is distributive

• Autorzy: Mikhail V. Volkov
• Języki publikacji: EN

Abstrakty

EN

We prove a theorem (for arbitrary ring varieties and, in a stronger form, for varieties of associative rings) which basically reduces the problem of a description of varieties with distributive subvariety lattice to the case of algebras over a finite prime field.

Słowa kluczowe

EN

variety of rings; subvariety lattice; distributive lattice; torsion-bounded variety; Mal'tsev product

Kategorie tematyczne

• 08B15: Lattices of varieties
• 06D05: Structure and representation theory

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2010
• Tom: 30
• Numer: 1
• Strony: 119-132

Daty

wydano

2010

otrzymano

2009-05-01

poprawiono

2009-11-04

Twórcy

autor

Mikhail V. Volkov

• Faculty of Mathematics and Mechanics, Ural State University, Lenina 51, 620083 Ekaterinburg, Russia

Bibliografia

• [1] D.S. Ananichev, Almost distributive varieties of Lie rings, Mat. Sbornik 186 (4) (1995), 3-20 [Russian; Engl. translation Sbornik: Math. 186 (4) (1995), 465-483].
• [2] V.A. Artamonov, On chain varieties of linear algebras, Trans. Am. Math. Soc. 221 (1976), 323-338. doi: 10.1090/S0002-9947-1976-0409572-5
• [3] V.A. Artamonov, Lattices of varieties of linear algebras, Uspekhi Mat. Nauk 33 (2) (1978) 135-167 [Russian; Engl. translation Russ. Math. Surv. 33 (2) (1978), 155-193].
• [4] G. Grätzer, General lattice theory, 2nd ed., Birkhäuser Verlag, Basel 1998.
• [5] A.I. Mal'tsev, On a multiplication of classes of algebraic systems, Sib. Mat. Zh. 8 (2) (1967), 346-365 [Russian; Engl. translation Sib. Math. J. 8 (1967), 254-267]. doi: 10.1007/BF02302476
• [6] Yu.N. Mal'tsev, On distributive varieties of associative algebras, in Investigations in the Theory of Rings, Algebras and Modules (Mat. Issledovaniya, Kishinev 76) (1984), 73-98 [Russian].
• [7] M.V. Volkov, Lattices of varieties of algebras, Mat. Sbornik 109 (1) (1979) 60-79 [Russian; Engl. translation Math. USSR, Sbornik 37 (1) (1980), 53-69].
• [8] M.V. Volkov, Periodic varieties of associative rings, Izvestiya VUZ. Matematika no.8 (1979) 3-13 [Russian; Engl. translation Soviet Math., Izv. VUZ 23 (8) (1979), 1-12].
• [9] M.V. Volkov, Identities in lattices of ring varieties, Algebra Universalis 23 (1986), 32-43. doi: 10.1007/BF01190909
• [10] M.V. Volkov and A.G. Geĭn, Identities of almost nilpotent Lie rings, Mat. Sbornik 118 (1) (1982), 132-142 [Russian; Engl. translation Math. USSR, Sbornik 46 (1) (1983), 133-142].
• [11] E.I. Zel'manov, On Engel Lie algebras, Sibirsk. Mat. Zh 26 (5) (1988), 112-117 [Russian; Engl. translation Siberian Math. J. 29 (5) (1988), 777-781].
• [12] K.A. Zhevlakov, A.M. Slinko, I.P. Shestakov and A.I. Shirshov, Rings that are nearly associative, Academic Press, New York 1982.

Identyfikatory

DOI

10.7151/dmgaa.1165