Centers in domains with quadratic growth

• Autorzy: Agata Smoktunowicz
• Języki publikacji: EN

Abstrakty

EN

Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.

Słowa kluczowe

EN

Centers; domains; growth of algebras; the Gelfand-Kirillov dimension; 16D90; 16P40; 16S80; 16W50

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2005
• Tom: 3
• Numer: 4
• Strony: 644-653

Daty

wydano

2005-12-01

online

2005-12-01

Twórcy

autor

Agata Smoktunowicz

• Polish Academy of Sciences

Bibliografia

• [1] M. Artin, W. Schelter and J. Tate: “The centers of 3-dimensional Skylyanian algebras”, In: Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math., Vol. 15, Academic Press, San Diego, CA, 1994, pp. 1–10.
• [2] M. Artin and J.T. Stafford: “Noncommutative graded domains with quadratic growth”, Invent. Math., Vol. 122(2), (1995), pp. 231–276. http://dx.doi.org/10.1007/BF01231444
• [3] J.P. Bell: private communication.
• [4] J.P. Bell and L.W. Small: “Centralizers in domains of Gelfand-Kirillov dimension 2”, Bull. London Math. Soc., Vol. 36(6), pp. 779–785.
• [5] G.M. Bergman: A note of growth functions of algebras and semigroups, mimeographed notes, University of California, Berkeley, 1978.
• [6] G. Krause and T. Lenagan: Growth of Algebras and Gelfand-Kirillov Dimension, Revised Edition, Graduate Studies in Mathematics, Vol. 22, American Society, Providence, 2000.
• [7] M. Lothaire, Algebraic Combinatorics of Words, Cambridge University Press 2002.
• [8] J.C. McConnel and J.C. Robson: Noncommutative Noetherian Rings, Wiley Interscience, Chichester, 1987.
• [9] L.W. Small, J.T. Stafford and R.B. Warfield Jr: “Affine algebras of Gelfand-Kirillov dimension one are PI”, Math. Proc. Cambridge Phil. Soc., Vol. 97, (1984), pp. 407–414. http://dx.doi.org/10.1017/S0305004100062976
• [10] L.W. Small and R.B. Warfield, Jr: “Prime affine algebras of Gelfand-Kirillov dimension one”, J. Algebra, Vol. 91, (1984) pp. 384–389. http://dx.doi.org/10.1016/0021-8693(84)90110-8
• [11] S.P. Smith and J.J. Zhang: “A remark of Gelfand-Kirillov dimension”, Proc. Amer. Math. Soc., Vol. 126(2), (1998), pp. 349–352. http://dx.doi.org/10.1090/S0002-9939-98-04074-X
• [12] A. Smoktunowicz: “On structure of domains with quadratic growth”, J. Algebra, Vol. 289(2), (2005), pp. 365–379. http://dx.doi.org/10.1016/j.jalgebra.2005.04.004
• [13] J.T. Stafford and M. Van den Bergh: “Noncommutative curves and noncommutative surfaces”, Bull. Am. Math. Soc., Vol. 38(2), pp. 171–216.
• [14] J.J. Zhang: “On lower transcendence degree”, Adv. Math., Vol. 139, (1998), pp. 157–193. http://dx.doi.org/10.1006/aima.1998.1749

Identyfikatory

DOI

10.2478/BF02475624