Abstract
Let A be an associative algebra over a field of characteristic zero. Then either all codimensions gc n (A) of its generalized polynomial identities are infinite or A is the sum of ideals I and J such that dim F I < ∞ and J is nilpotent. In the latter case, there exist numbers n 0 ∈ ℕ, C ∈ ℚ+, and t ∈ ℤ+ for which gc n (A) < +∞ if n ≥ n 0 and gc n (A) ∼ Cn t d n as n → ∞, where d = PIexp(A) ∈ ℤ+. Thus, in the latter case, conjectures of Amitsur and Regev on generalized codimensions hold.
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References
A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, in Math. Surveys Monogr. (Amer.Math. Soc., Providence, RI, 2005), Vol. 122.
A. Berele and A. Regev, “Asymptotic behaviour of codimensions of p.i. algebras satisfying Capelli identities,” Trans. Amer. Math. Soc. 360(10), 5155–5172 (2008).
V. Drensky, “Relations for the cocharacter sequences of T-ideals,” in Proceedings of the International Conference on Algebra Dedicated to the Memory of A. I. Mal’cev (Novosibirsk, 1989), Pt. 2, pp. 285–300 [in Contemp. Math. (Amer. Math. Soc., Providence, RI, 1992), Vol. 131, Pt. 2, pp. 285–300].
A. S. Gordienko, “The Regev conjecture and cocharacters for identities of associative algebras of the PIexponents 1 and 2,” Mat. Zametki 83(6), 815–824 (2008) [Math. Notes 83 (5–6), 744–752 (2008)].
K. I. Beidar, W. S. Martindale, III, and A. V. Mikhalev, Rings with Generalized Identities, in Monogr. Textbooks Pure Appl. Math. (Marcel Dekker, New York, NY, 1996), Vol. 196.
A. S. Gordienko, Mat. Sb. (to appear).
A. Regev, “Codimensions and trace codimensions of matrices are asymptotically equal,” Israel J. Math. 47(2–3), 246–250 (1984).
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Original Russian Text © A. S. Gordienko, 2009, published in Matematicheskie Zametki, 2009, Vol. 86, No. 5, pp. 681–685.
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Gordienko, A.S. A finiteness criterion and asymptotics for codimensions of generalized identities. Math Notes 86, 645–649 (2009). https://doi.org/10.1134/S0001434609110066
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DOI: https://doi.org/10.1134/S0001434609110066