A reduction theorem for ring varieties whose subvariety lattice is distributive

Mikhail V. Volkov

Discussiones Mathematicae - General Algebra and Applications (2010)

  • Volume: 30, Issue: 1, page 119-132
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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Mikhail V. Volkov. "A reduction theorem for ring varieties whose subvariety lattice is distributive." Discussiones Mathematicae - General Algebra and Applications 30.1 (2010): 119-132. <http://eudml.org/doc/276650>.

@article{MikhailV2010,
abstract = {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.},
author = {Mikhail V. Volkov},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {variety of rings; subvariety lattice; distributive lattice; torsion-bounded variety; Mal'tsev product},
language = {eng},
number = {1},
pages = {119-132},
title = {A reduction theorem for ring varieties whose subvariety lattice is distributive},
url = {http://eudml.org/doc/276650},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Mikhail V. Volkov
TI - A reduction theorem for ring varieties whose subvariety lattice is distributive
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2010
VL - 30
IS - 1
SP - 119
EP - 132
AB - 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.
LA - eng
KW - variety of rings; subvariety lattice; distributive lattice; torsion-bounded variety; Mal'tsev product
UR - http://eudml.org/doc/276650
ER -

References

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  1. [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]. Zbl0841.17006
  2. [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 Zbl0337.17001
  3. [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. [4] G. Grätzer, General lattice theory, 2nd ed., Birkhäuser Verlag, Basel 1998. Zbl0909.06002
  5. [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. [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]. Zbl0559.16011
  7. [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]. Zbl0411.17008
  8. [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. [9] M.V. Volkov, Identities in lattices of ring varieties, Algebra Universalis 23 (1986), 32-43. doi: 10.1007/BF01190909 Zbl0607.17001
  10. [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]. Zbl0494.17009
  11. [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. [12] K.A. Zhevlakov, A.M. Slinko, I.P. Shestakov and A.I. Shirshov, Rings that are nearly associative, Academic Press, New York 1982. Zbl0487.17001

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