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Fundam. Prikl. Mat., 1999, Volume 5, Issue 1, Pages 47–66 (Mi fpm365)  

This article is cited in 29 scientific papers (total in 29 papers)

On non-Spechtian varieties

A. Ya. Belov

House of scientific and technical work of youth

Abstract: This article is devoted to construction of infinitely based series of identities. Such counterexamples in Specht problem are built in any positive characteristics. The main result is the following:
Theorem. Let $F$ be any field of characteristic $p$, $q=p^s$, $s>1$. Then the polynomials $R_n$:
$$ R_n=[[E,T],T]\prod_{i=1}^n Q(x_i,y_i) ([T,[T,F]][[E,T],T])^{q-1}[T,[T,F]], $$
where $Q(x,y)=x^{p-1}y^{p-1}[x,y]$, generate an infinitely based variety.

Full text: PDF file (1053 kB)

Bibliographic databases:
UDC: 512.55
Received: 01.11.1998

Citation: A. Ya. Belov, “On non-Spechtian varieties”, Fundam. Prikl. Mat., 5:1 (1999), 47–66

Citation in format AMSBIB
\Bibitem{Bel99}
\by A.~Ya.~Belov
\paper On non-Spechtian varieties
\jour Fundam. Prikl. Mat.
\yr 1999
\vol 5
\issue 1
\pages 47--66
\mathnet{http://mi.mathnet.ru/fpm365}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1799544}
\zmath{https://zbmath.org/?q=an:0964.16024}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Shchigolev, “The finite basis property of $T$-spaces over fields of characteristic zero”, Izv. Math., 65:5 (2001), 1041–1071  mathnet  crossref  crossref  mathscinet  zmath
    2. Giambruno, A, “On the identities of the Grassmann algebras in characteristic p > 0”, Israel Journal of Mathematics, 122 (2001), 305  crossref  mathscinet  zmath  isi
    3. Gupta, CK, “A simple example of a non-finitely based system of polynomial identities”, Communications in Algebra, 30:10 (2002), 4851  crossref  mathscinet  zmath  isi
    4. Gupta, CK, “A non-finitely based system of polynomial identities which contains the identity x(6)=0”, Quarterly Journal of Mathematics, 53 (2002), 173  crossref  mathscinet  isi
    5. A. Ya. Belov, “No associative $PI$-algebra coincides with its commutant”, Siberian Math. J., 44:6 (2003), 969–980  mathnet  crossref  mathscinet  zmath  isi  elib
    6. Gupta, CK, “The finite basis question for varieties of groups - Some recent results”, Illinois Journal of Mathematics, 47:1–2 (2003), 273  mathscinet  zmath  isi
    7. A. V. Grishin, “Model algebras, multiplicities, and representability indices of varieties of associative algebras”, Sb. Math., 195:1 (2004), 1–18  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. Drensky V., “Polynomial identity rings - Part A - Combinatorial aspects in PI-rings”, Polynomial Identity Rings, Advanced Courses in Mathematics CRM Barcelona, 2004, 1  mathscinet  zmath  isi
    9. E. A. Kireeva, A. N. Krasilnikov, “On Some Extremal Varieties of Associative Algebras”, Math. Notes, 78:4 (2005), 503–517  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. Crvenkovic, S, “A locally finite variety of rings with an undecidable equational theory”, Quarterly Journal of Mathematics, 57 (2006), 297  crossref  mathscinet  zmath  isi
    11. Drensky, V, “Grobner bases of ideals invariant under endomorphisms”, Journal of Symbolic Computation, 41:7 (2006), 835  crossref  mathscinet  zmath  isi
    12. A. Ya. Belov, “On Rings Asymptotically Close to Associative Rings”, Siberian Adv. Math., 17:4 (2007), 227–267  mathnet  crossref  mathscinet  elib
    13. E. A. Kireeva, “Limit T-spaces”, J. Math. Sci., 152:4 (2008), 540–557  mathnet  crossref  mathscinet  zmath
    14. L. M. Tsybulya, “Theorems on equalization and monomiality in a relatively free Grassmann algebra”, J. Math. Sci., 163:6 (2009), 759–773  mathnet  crossref  mathscinet  elib  elib
    15. A. V. Grishin, L. M. Tsybulya, “On the multiplicative and $T$-space structure of the relatively free Grassmann algebra”, Sb. Math., 200:9 (2009), 1299–1338  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. Aladova E.V., Krasilnikov A.N., “Polynomial Identities in Nil-Algebras”, Transactions of the American Mathematical Society, 361:11 (2009), 5629–5646  crossref  mathscinet  zmath  isi
    17. A. V. Grishin, L. M. Tsybulya, “On the structure of a relatively free Grassmann algebra”, J. Math. Sci., 171:2 (2010), 149–212  mathnet  crossref  mathscinet  elib
    18. 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
    19. Silkin N., “On Finite Basis Property for Joins of Varieties of Associative Rings”, Comm Algebra, 38:9 (2010), 3187–3205  crossref  mathscinet  zmath  isi  elib
    20. Klyachko A.A., Menshova E.V., “The identities of additive binary arithmetics”, Electron J Combin, 19:1 (2012), P40  mathscinet  zmath  isi
    21. Goncalves D.J., Krasilnikov A., Sviridova I., “Limit T-Subspaces and the Central Polynomials in N Variables of the Grassmann Algebra”, J. Algebra, 371 (2012), 156–174  crossref  mathscinet  zmath  isi  elib
    22. Edmond W. H. Lee, “Maximal Specht varieties of monoids”, Mosc. Math. J., 12:4 (2012), 787–802  mathnet  mathscinet
    23. G. S. Deryabina, A. N. Krasilnikov, “A Non-Finitely-Based Variety of Centrally Metabelian Pointed Groups”, Math. Notes, 95:5 (2014), 743–746  mathnet  crossref  crossref  mathscinet  isi  elib
    24. Concalves D.J., Krasilnikov A., Sviridova I., “Limit T-Subalgebras in Free Associative Algebras”, J. Algebra, 412 (2014), 264–280  crossref  mathscinet  isi
    25. M. I. Kharitonov, “Otsenki, svyazannye s teoremoi Shirshova o vysote”, Chebyshevskii sb., 15:4 (2014), 55–123  mathnet
    26. Deryabina G., Krasilnikov A., “the Subalgebra of Graded Central Polynomials of An Associative Algebra”, J. Algebra, 425 (2015), 313–323  crossref  mathscinet  zmath  isi  elib
    27. M. I. Kharitonov, “The estimate of the number of permutationally-ordered sets”, Moscow University Mathematics Bulletin, 70:3 (2015), 125–129  mathnet  crossref  mathscinet
    28. Di Vincenzo O.M., Koshlukov P., Tomaz da Silva V.R., “On Z(P)-Graded Identities and Cocharacters of the Grassmann Algebra”, Commun. Algebr., 45:1 (2017), 343–356  crossref  mathscinet  zmath  isi  scopus
    29. Kireeva E., Shchigolev V., “The Nilpotence Degree of Quantum Lie Nilpotent Algebras”, Int. J. Algebr. Comput., 28:6 (2018), 1119–1128  crossref  mathscinet  zmath  isi  scopus
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