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 Uspekhi Mat. Nauk, 1969, Volume 24, Issue 1(145), Pages 39–42 (Mi umn5449)

Series of articles on the multioperator rings and algebras
Two theorems on identities in multioperator algebras

F. I. Kizner

Abstract: Two (unconnected) propositions on $\Omega$-algebras with identical relations are proved. The first of these (Theorem 1, in § 1) generalizes to $\Omega$-algebras a known fact from the theory of associative linear algebras, which asserts that every finite-dimensional algebra is an algebra with identical relations (more exactly, every algebra $A$ of dimension over a field $m$ satisfies a so-called standard identity of degree $m+1$). In § 2 we prove that every identical relation in an $\Omega$-algebra over a field of characteristic zero is equivalent to a system of polylinear identical relations (Theorem 2), from which it follows that the study of $\Omega$-algebras with arbitrary identical relations reduces to that of $\Omega$-algebras with polylinear identical relations. This theorem is proved in practically the same way as the corresponding proposition for ordinary algebras with identical relations, that is, algebras with a single binary multiplication (see for example, Mal'tsev [1]); it is clearly a generalization of it.

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English version:
Russian Mathematical Surveys, 1969, 24:1, 37–40

Bibliographic databases:

UDC: 519.4+519.9
MSC: 16R10, 47C05

Citation: F. I. Kizner, “Two theorems on identities in multioperator algebras”, Uspekhi Mat. Nauk, 24:1(145) (1969), 39–42; Russian Math. Surveys, 24:1 (1969), 37–40

Citation in format AMSBIB
\Bibitem{Kiz69} \by F.~I.~Kizner \paper Two theorems on identities in multioperator algebras \jour Uspekhi Mat. Nauk \yr 1969 \vol 24 \issue 1(145) \pages 39--42 \mathnet{http://mi.mathnet.ru/umn5449} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=237406} \zmath{https://zbmath.org/?q=an:0191.03202|0211.06302} \transl \jour Russian Math. Surveys \yr 1969 \vol 24 \issue 1 \pages 37--40 \crossref{https://doi.org/10.1070/RM1969v024n01ABEH001337} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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, V. A. Artamonov, “Some properties of subalgebras in varieties of linear $\Omega$-albebras”, Math. USSR-Sb., 16:1 (1972), 69–85
5. T. M. Baranovich, M. S. Burgin, “Linear $\Omega$-algebras”, Russian Math. Surveys, 30:4 (1975), 65–113
6. S. N. Tronin, “Operads and varieties of algebras defined by polylinear identities”, Siberian Math. J., 47:3 (2006), 555–573
7. S. N. Tronin, “Superalgebras and operads. I”, Siberian Math. J., 50:3 (2009), 503–514
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