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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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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
2. Giambruno, A, “On the identities of the Grassmann algebras in characteristic p > 0”, Israel Journal of Mathematics, 122 (2001), 305
3. Gupta, CK, “A simple example of a non-finitely based system of polynomial identities”, Communications in Algebra, 30:10 (2002), 4851
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
5. A. Ya. Belov, “No associative $PI$-algebra coincides with its commutant”, Siberian Math. J., 44:6 (2003), 969–980
6. Gupta, CK, “The finite basis question for varieties of groups - Some recent results”, Illinois Journal of Mathematics, 47:1–2 (2003), 273
7. A. V. Grishin, “Model algebras, multiplicities, and representability indices of varieties of associative algebras”, Sb. Math., 195:1 (2004), 1–18
8. Drensky V., “Polynomial identity rings - Part A - Combinatorial aspects in PI-rings”, Polynomial Identity Rings, Advanced Courses in Mathematics CRM Barcelona, 2004, 1
9. E. A. Kireeva, A. N. Krasilnikov, “On Some Extremal Varieties of Associative Algebras”, Math. Notes, 78:4 (2005), 503–517
10. Crvenkovic, S, “A locally finite variety of rings with an undecidable equational theory”, Quarterly Journal of Mathematics, 57 (2006), 297
11. Drensky, V, “Grobner bases of ideals invariant under endomorphisms”, Journal of Symbolic Computation, 41:7 (2006), 835
12. A. Ya. Belov, “On Rings Asymptotically Close to Associative Rings”, Siberian Adv. Math., 17:4 (2007), 227–267
13. E. A. Kireeva, “Limit T-spaces”, J. Math. Sci., 152:4 (2008), 540–557
14. L. M. Tsybulya, “Theorems on equalization and monomiality in a relatively free Grassmann algebra”, J. Math. Sci., 163:6 (2009), 759–773
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
16. Aladova E.V., Krasilnikov A.N., “Polynomial Identities in Nil-Algebras”, Transactions of the American Mathematical Society, 361:11 (2009), 5629–5646
17. A. V. Grishin, L. M. Tsybulya, “On the structure of a relatively free Grassmann algebra”, J. Math. Sci., 171:2 (2010), 149–212
18. A. Ya. Belov, “The local finite basis property and local representability of varieties of associative rings”, Izv. Math., 74:1 (2010), 1–126
19. Silkin N., “On Finite Basis Property for Joins of Varieties of Associative Rings”, Comm Algebra, 38:9 (2010), 3187–3205
20. Klyachko A.A., Menshova E.V., “The identities of additive binary arithmetics”, Electron J Combin, 19:1 (2012), P40
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
22. Edmond W. H. Lee, “Maximal Specht varieties of monoids”, Mosc. Math. J., 12:4 (2012), 787–802
23. G. S. Deryabina, A. N. Krasilnikov, “A Non-Finitely-Based Variety of Centrally Metabelian Pointed Groups”, Math. Notes, 95:5 (2014), 743–746
24. Concalves D.J., Krasilnikov A., Sviridova I., “Limit T-Subalgebras in Free Associative Algebras”, J. Algebra, 412 (2014), 264–280
25. M. I. Kharitonov, “Otsenki, svyazannye s teoremoi Shirshova o vysote”, Chebyshevskii sb., 15:4 (2014), 55–123
26. Deryabina G., Krasilnikov A., “the Subalgebra of Graded Central Polynomials of An Associative Algebra”, J. Algebra, 425 (2015), 313–323
27. M. I. Kharitonov, “The estimate of the number of permutationally-ordered sets”, Moscow University Mathematics Bulletin, 70:3 (2015), 125–129
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
29. Kireeva E., Shchigolev V., “The Nilpotence Degree of Quantum Lie Nilpotent Algebras”, Int. J. Algebr. Comput., 28:6 (2018), 1119–1128
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