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Uspekhi Mat. Nauk, 1969, Volume 24, Issue 1(145), Pages 27–38 (Mi umn5448)  

This article is cited in 8 scientific papers (total in 8 papers)


Series of articles on the multioperator rings and algebras
The freeness theorem in some varieties of linear algebras and rings

M. S. Burgin


Abstract: In this paper we study properties of free and near-free linear $\Omega$-algebras (over a field) lying in a variety $\mathfrak{M}_P$ given by permutational identities. Examples of sub-identities in the case of ordinary linear algebras (with a single binary operation) are the commutative and anticommutative laws. The identities, studied by Polin in [4] are also special cases of identities of this kind. The auxiliary results derived in the first two sections yield a method of proof for varieties $\mathfrak{M}_P$ of the freeness theorem analoguos to the Dehn-Magnus theorem for groups [7], Zhukov's theorem for non-associative algebras [2], and Shirshov's theorems for commutative and anticommutative algebras [5] and Lie algebras [6]. Zhukov' s theorem [2] and Shirshov's theorem [5] are special cases of our proposition. We note that although generally speaking a subalgebra of a free algebra in $\mathfrak{M}_P$ need not be free in the variety, the freeness theorem is always true for such varieties. It is known that for non-associative rings, in contrast to the case of linear algebras, the theorem on subrings of a free ring is false in the most general case. However, using the comparison of an $\Omega$-ring with a linear $\Omega$-algebra over the rational field, we obtain in § 3 a freeness theorem for $\Omega$-rings. The author expresses his indebtedness to A. G. Kurosh for valuable advice and remarks during the progress of the work, and for help in preparing the manuscript for the printer.

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English version:
Russian Mathematical Surveys, 1969, 24:1, 25–35

Bibliographic databases:

UDC: 519.4+519.9
MSC: 17A50, 17A30, 08A30
Received: 30.09.1968

Citation: M. S. Burgin, “The freeness theorem in some varieties of linear algebras and rings”, Uspekhi Mat. Nauk, 24:1(145) (1969), 27–38; Russian Math. Surveys, 24:1 (1969), 25–35

Citation in format AMSBIB
\Bibitem{Bur69}
\by M.~S.~Burgin
\paper The freeness theorem in some varieties of linear algebras and rings
\jour Uspekhi Mat. Nauk
\yr 1969
\vol 24
\issue 1(145)
\pages 27--38
\mathnet{http://mi.mathnet.ru/umn5448}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=237405}
\zmath{https://zbmath.org/?q=an:0177.02701}
\transl
\jour Russian Math. Surveys
\yr 1969
\vol 24
\issue 1
\pages 25--35
\crossref{https://doi.org/10.1070/RM1969v024n01ABEH001336}


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    Erratum

    This publication is cited in the following articles:
    1. A. G. Kurosh, “Multioperator rings and algebras”, Russian Math. Surveys, 24:1 (1969), 1–13  mathnet  crossref  mathscinet  zmath
    2. M. S. Burgin, “Commutative products of linear $\Omega$-algebras”, Math. USSR-Izv., 4:5 (1970), 979–999  mathnet  crossref  mathscinet  zmath
    3. M. S. Burgin, “Gruppoid mnogoobrazii lineinykh $\Omega$-algebr”, UMN, 25:3(153) (1970), 263–264  mathnet  mathscinet  zmath
    4. M. S. Burgin, “Shreierovy mnogoobraziya lineinykh $\Omega$-algebr”, UMN, 27:5(167) (1972), 227–228  mathnet  mathscinet  zmath
    5. M. S. Burgin, V. A. Artamonov, “Some properties of subalgebras in varieties of linear $\Omega$-albebras”, Math. USSR-Sb., 16:1 (1972), 69–85  mathnet  crossref  mathscinet  zmath
    6. M. S. Burgin, “Schreier varieties of linear $\Omega$-algebras”, Math. USSR-Sb., 22:4 (1974), 561–579  mathnet  crossref  mathscinet  zmath
    7. T. M. Baranovich, M. S. Burgin, “Linear $\Omega$-algebras”, Russian Math. Surveys, 30:4 (1975), 65–113  mathnet  crossref  mathscinet  zmath
    8. V. A. Artamonov, A. V. Klimakov, A. A. Mikhalev, A. V. Mikhalev, “Primitive and almost primitive elements of Schreier varieties”, J. Math. Sci., 237:2 (2019), 157–179  mathnet  crossref  elib
  • Успехи математических наук Russian Mathematical Surveys
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