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Algebra Logika, 2004, Volume 43, Number 4, Pages 482–505 (Mi al86)  

This article is cited in 9 scientific papers (total in 9 papers)

Varieties of Associative Algebras Satisfying Engel Identities

O. B. Finogenova

Ural State University

Abstract: A variety of associative algebras (rings) is said to be Engel if it satisfies an identity of the form $[\ldots[[x,y],y],\ldots,y]=0$. On the Zorn lemma, every non-Engel variety contains some just non-Engel variety, that is, a minimal (w.r.t. inclusion) element in the set of all non-Engel varieties. A list of such varieties for algebras over a field of characteristic 0 was made up by Yu. N. Mal'tsev. Here, we present a complete description of just non-Engel varieties both for the case of algebras over a field of positive characteristic and for the case of rings. This gives the answer to Question 3.53 in the Dniester Notebook.

Keywords: Engel identity, just non-Engel variety, variety of associative rings, associative algebra over a field

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English version:
Algebra and Logic, 2004, 43:4, 271–284

Bibliographic databases:

UDC: 512.552.4
Received: 22.04.2003

Citation: O. B. Finogenova, “Varieties of Associative Algebras Satisfying Engel Identities”, Algebra Logika, 43:4 (2004), 482–505; Algebra and Logic, 43:4 (2004), 271–284

Citation in format AMSBIB
\Bibitem{Fin04}
\by O.~B.~Finogenova
\paper Varieties of Associative Algebras Satisfying Engel Identities
\jour Algebra Logika
\yr 2004
\vol 43
\issue 4
\pages 482--505
\mathnet{http://mi.mathnet.ru/al86}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2105850}
\zmath{https://zbmath.org/?q=an:1065.16019}
\transl
\jour Algebra and Logic
\yr 2004
\vol 43
\issue 4
\pages 271--284
\crossref{https://doi.org/10.1023/B:ALLO.0000035118.51742.41}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-42249093859}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. R. Chekhlov, “Torsion-Free Weakly Transitive $E$-Engel Abelian Groups”, Math. Notes, 94:4 (2013), 583–589  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. O. B. Finogenova, “Almost commutative varieties of associative rings and algebras over a finite field”, Algebra and Logic, 52:6 (2014), 484–510  mathnet  crossref  mathscinet  isi
    3. O. B. Finogenova, “Pochti lievo nilpotentnye mnogoobraziya assotsiativnykh kolets”, Sib. elektron. matem. izv., 12 (2015), 901–909  mathnet  crossref
    4. O. B. Finogenova, “Pochti lievo nilpotentnye nepervichnye mnogoobraziya assotsiativnykh algebr”, Tr. IMM UrO RAN, 21, no. 4, 2015, 282–291  mathnet  mathscinet  elib
    5. Jespers E., Riley D., Shahada M., “Multiplicatively Collapsing and Rewritable Algebras”, Proc. Amer. Math. Soc., 143:10 (2015), PII S0002-9939(2015)12563-4, 4223–4236  crossref  mathscinet  zmath  isi  elib  scopus
    6. Finogenova O.B., “Varieties Satisfying Semigroup Identities: Algebras Over a Finite Field and Rings”, Int. J. Algebr. Comput., 26:5 (2016), 985–1017  crossref  mathscinet  zmath  isi  scopus
    7. Riley D.M., “When Is a Power of the Frobenius Map on a Noncommutative Ring a Homomorphism?”, J. Algebra, 479 (2017), 159–172  crossref  mathscinet  zmath  isi  scopus
    8. A. V. Kislitsin, “On nonnilpotent almost commutative $L$-varieties of vector spaces”, Siberian Math. J., 59:3 (2018), 458–462  mathnet  crossref  crossref  isi  elib
    9. A. V. Kislitsin, “The Specht property of $L$-varieties of vector spaces over an arbitrary field”, Algebra and Logic, 57:5 (2018), 360–367  mathnet  crossref  crossref  isi
  • Алгебра и логика Algebra and Logic
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