Centers in domains with quadratic growth

Agata Smoktunowicz

Open Mathematics (2005)

  • Volume: 3, Issue: 4, page 644-653
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Agata Smoktunowicz. "Centers in domains with quadratic growth." Open Mathematics 3.4 (2005): 644-653. <http://eudml.org/doc/268849>.

@article{AgataSmoktunowicz2005,
abstract = {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.},
author = {Agata Smoktunowicz},
journal = {Open Mathematics},
keywords = {16D90; 16P40; 16S80; 16W50},
language = {eng},
number = {4},
pages = {644-653},
title = {Centers in domains with quadratic growth},
url = {http://eudml.org/doc/268849},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Agata Smoktunowicz
TI - Centers in domains with quadratic growth
JO - Open Mathematics
PY - 2005
VL - 3
IS - 4
SP - 644
EP - 653
AB - 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.
LA - eng
KW - 16D90; 16P40; 16S80; 16W50
UR - http://eudml.org/doc/268849
ER -

References

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  1. [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. Zbl0823.17019
  2. [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 Zbl0849.16022
  3. [3] J.P. Bell: private communication. 
  4. [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. Zbl1080.16015
  5. [5] G.M. Bergman: A note of growth functions of algebras and semigroups, mimeographed notes, University of California, Berkeley, 1978. 
  6. [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. Zbl0957.16001
  7. [7] M. Lothaire, Algebraic Combinatorics of Words, Cambridge University Press 2002. Zbl1001.68093
  8. [8] J.C. McConnel and J.C. Robson: Noncommutative Noetherian Rings, Wiley Interscience, Chichester, 1987. 
  9. [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 Zbl0561.16005
  10. [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 Zbl0545.16011
  11. [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 Zbl0896.16019
  12. [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 Zbl1079.16010
  13. [13] J.T. Stafford and M. Van den Bergh: “Noncommutative curves and noncommutative surfaces”, Bull. Am. Math. Soc., Vol. 38(2), pp. 171–216. Zbl1042.16016
  14. [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 Zbl0924.16015

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