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 Mat. Sb. (N.S.), 1988, Volume 135(177), Number 3, Pages 373–384 (Mi msb1708)

On a Shirshov basis of relatively free algebras of complexity $n$

A. Ya. Belov

Abstract: A Shirshov basis is a set of elements of an algebra $A$ over which $A$ has bounded height in the sense of Shirshov.
A description is given of Shirshov bases consisting of words for associative or alternative relatively free algebras over an arbitrary commutative associative ring $\Phi$ with unity. It is proved that the set of monomials of degree at most $m^2$ is a Shirshov basis in a Jordan PI-algebra of degree $m$. It is shown that under certain conditions on $\operatorname{var}(B)$ (satisfied by alternative and Jordan PI-algebras), if each factor of $B$ with nilpotent projections of all elements of $M$ is nilpotent, then $M$ is a Shirshov basis of $B$ if $M$ generates $B$ as an algebra.
Bibliography: 12 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 63:2, 363–374

Bibliographic databases:

UDC: 519.48
MSC: Primary 16A06, 17D05; Secondary 16A38

Citation: A. Ya. Belov, “On a Shirshov basis of relatively free algebras of complexity $n$”, Mat. Sb. (N.S.), 135(177):3 (1988), 373–384; Math. USSR-Sb., 63:2 (1989), 363–374

Citation in format AMSBIB
\Bibitem{Bel88} \by A.~Ya.~Belov \paper On a~Shirshov basis of relatively free algebras of complexity~$n$ \jour Mat. Sb. (N.S.) \yr 1988 \vol 135(177) \issue 3 \pages 373--384 \mathnet{http://mi.mathnet.ru/msb1708} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=937647} \zmath{https://zbmath.org/?q=an:0667.16015|0659.16012} \transl \jour Math. USSR-Sb. \yr 1989 \vol 63 \issue 2 \pages 363--374 \crossref{https://doi.org/10.1070/SM1989v063n02ABEH003279} 

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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. Belov A., “About Height Theorem”, Commun. Algebr., 23:9 (1995), 3551–3553
2. Drensky V., “Polynomial identity rings - Part A - Combinatorial aspects in PI-rings”, Polynomial Identity Rings, Advanced Courses in Mathematics Crm Barcelona, 2004, 1
3. Kanel-Belov A., Rowen L.H., Vishne U., “Normal Bases of Pi-Algebras”, Adv. Appl. Math., 37:3 (2006), 378–389
4. A. Ya. Belov, “The Kurosh problem, height theorem, nilpotency of the radical, and algebraicity identity”, J. Math. Sci., 154:2 (2008), 125–142
5. A. Ya. Belov, “Burnside-type problems, theorems on height, and independence”, J. Math. Sci., 156:2 (2009), 219–260
6. A. Ya. Belov, “On Rings Asymptotically Close to Associative Rings”, Siberian Adv. Math., 17:4 (2007), 227–267
7. Letzner E.S., “Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension”, J. Algebra, 320:11 (2008), 3926–3934
8. A. Ya. Belov, “The local finite basis property and local representability of varieties of associative rings”, Izv. Math., 74:1 (2010), 1–126
9. A. Ya. Belov, M. I. Kharitonov, “Subexponential estimates in Shirshov's theorem on height”, Sb. Math., 203:4 (2012), 534–553
10. A. Ya. Belov, M. I. Kharitonov, “Subexponential estimates in the height theorem and estimates on numbers of periodic parts of small periods”, J. Math. Sci., 193:4 (2013), 493–515
11. Vesselin Drensky, Şehmus F{\i}nd{\i}k, “Inner automorphisms of Lie algebras related with generic $2\times 2$ matrices”, Algebra Discrete Math., 14:1 (2012), 49–70
12. M. I. Kharitonov, “Otsenki, svyazannye s teoremoi Shirshova o vysote”, Chebyshevskii sb., 15:4 (2014), 55–123
13. KanelBelov A. Karasik Y. Rowen L., “Computational Aspects of Polynomial Identities: Vol 1, Kemer'S Theorems, 2Nd Edition”, Computational Aspects of Polynomial Identities: Vol 1, Kemer'S Theorems, 2Nd Edition, Monographs and Research Notes in Mathematics, 16, Crc Press-Taylor & Francis Group, 2016, 1–407
14. Pchelintsev S.V., “Proper identities of finitely generated commutative alternative algebras”, J. Algebra, 470 (2017), 425–440
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