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

This article is cited in 14 scientific papers (total in 14 papers)

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
Received: 06.10.1986

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
\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
\jour Math. USSR-Sb.
\yr 1989
\vol 63
\issue 2
\pages 363--374

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    This publication is cited in the following articles:
    1. Belov A., “About Height Theorem”, Commun. Algebr., 23:9 (1995), 3551–3553  crossref  mathscinet  zmath  isi
    2. Drensky V., “Polynomial identity rings - Part A - Combinatorial aspects in PI-rings”, Polynomial Identity Rings, Advanced Courses in Mathematics Crm Barcelona, 2004, 1  isi
    3. Kanel-Belov A., Rowen L.H., Vishne U., “Normal Bases of Pi-Algebras”, Adv. Appl. Math., 37:3 (2006), 378–389  crossref  mathscinet  zmath  isi
    4. A. Ya. Belov, “The Kurosh problem, height theorem, nilpotency of the radical, and algebraicity identity”, J. Math. Sci., 154:2 (2008), 125–142  mathnet  crossref  mathscinet  zmath  elib  elib
    5. A. Ya. Belov, “Burnside-type problems, theorems on height, and independence”, J. Math. Sci., 156:2 (2009), 219–260  mathnet  crossref  mathscinet  zmath  elib  elib
    6. A. Ya. Belov, “On Rings Asymptotically Close to Associative Rings”, Siberian Adv. Math., 17:4 (2007), 227–267  mathnet  crossref  mathscinet  elib
    7. Letzner E.S., “Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension”, J. Algebra, 320:11 (2008), 3926–3934  crossref  mathscinet  isi
    8. A. Ya. Belov, “The local finite basis property and local representability of varieties of associative rings”, Izv. Math., 74:1 (2010), 1–126  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. A. Ya. Belov, M. I. Kharitonov, “Subexponential estimates in Shirshov's theorem on height”, Sb. Math., 203:4 (2012), 534–553  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref
    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  mathnet  mathscinet  zmath
    12. M. I. Kharitonov, “Otsenki, svyazannye s teoremoi Shirshova o vysote”, Chebyshevskii sb., 15:4 (2014), 55–123  mathnet
    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  mathscinet  isi
    14. Pchelintsev S.V., “Proper identities of finitely generated commutative alternative algebras”, J. Algebra, 470 (2017), 425–440  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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