Abstract:
The subject of this work is an extension of A. R. Kemer's results to a rather broad class of rings close to associative rings, over a field of characteristic 0 (in particular, this class includes the varieties generated by finite-dimensional alternative and Jordan rings). We prove the finite-basedness of systems of identities (the Specht property), the representability of finitely-generated relatively free algebras, and the rationality of their Hilbert series. For this purpose, we extend the Razymslov-Zubrilin theory to Kemer polynomials. For a rather broad class of varieties, we prove Shirshov's theorem on height.
Key words:
PI-algebra, representable algebra, universal algebra, nonassociative algebra, alternative algebra, Jordan algebra, signature, polynomial identity, Hilbert series, Specht problem.
\Bibitem{Bel07}
\by A.~Ya.~Belov
\paper On Rings Asymptotically Close to Associative Rings
\jour Mat. Tr.
\yr 2007
\vol 10
\issue 1
\pages 29--96
\mathnet{http://mi.mathnet.ru/mt29}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2485366}
\elib{https://elibrary.ru/item.asp?id=9483454}
\transl
\jour Siberian Adv. Math.
\yr 2007
\vol 17
\issue 4
\pages 227--267
\crossref{https://doi.org/10.3103/S1055134407040013}
Linking options:
https://www.mathnet.ru/eng/mt29
https://www.mathnet.ru/eng/mt/v10/i1/p29
This publication is cited in the following 5 articles:
Yang Zhang, Jizhu Nan, Yongsheng Ma, “A conjecture of Mallows and Sloane with the universal denominator of Hilbert series”, Open Mathematics, 22:1 (2024)
Vladimir Dotsenko, Nurlan Ismailov, Ualbai Umirbaev, “Polynomial identities in Novikov algebras”, Math. Z., 303:3 (2023)
Belov-Kanel A., Rowen L., Vishne U., “Specht'S Problem For Associative Affine Algebras Over Commutative Noetherian Rings”, Trans. Am. Math. Soc., 367:8 (2015), 5553–5596
A. Ya. Belov, “The local finite basis property and local representability of varieties of associative rings”, Izv. Math., 74:1 (2010), 1–126
A. V. Grishin, L. M. Tsybulya, “On the structure of a relatively free Grassmann algebra”, J. Math. Sci., 171:2 (2010), 149–212
Citation in format AMSBIB
Contact us: