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

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

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
2. M. S. Burgin, “Commutative products of linear $\Omega$-algebras”, Math. USSR-Izv., 4:5 (1970), 979–999
3. M. S. Burgin, “Gruppoid mnogoobrazii lineinykh $\Omega$-algebr”, UMN, 25:3(153) (1970), 263–264
4. M. S. Burgin, “Shreierovy mnogoobraziya lineinykh $\Omega$-algebr”, UMN, 27:5(167) (1972), 227–228
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
6. M. S. Burgin, “Schreier varieties of linear $\Omega$-algebras”, Math. USSR-Sb., 22:4 (1974), 561–579
7. T. M. Baranovich, M. S. Burgin, “Linear $\Omega$-algebras”, Russian Math. Surveys, 30:4 (1975), 65–113
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
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