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Primitive noetherian algebras with big centers

Author: Ronald S. Irving
Journal: Proc. Amer. Math. Soc. 129 (2001), 1587-1593
MSC (2000): Primary 16D60, 16P40; Secondary 16S36
Published electronically: October 31, 2000
MathSciNet review: 1814084
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Recent work of Artin, Small, and Zhang extends Grothendieck's classical commutative algebra result on generic freeness to a large family of non-commutative algebras. Over such an algebra, any finitely-generated module becomes free after localization at a suitable central element. In this paper, a construction is given of primitive noetherian algebras, finitely generated over the integers or over algebraic closures of finite fields, such that the faithful, simple modules don't satisfy such a freeness condition. These algebras also fail to satisfy a non-commutative version of the Nullstellensatz.

References [Enhancements On Off] (What's this?)

  • 1. M. Artin, L. W. Small, and J. J. Zhang, Generic flatness for strongly noetherian algebras, J. Algebra 221 (1999), 579-610. CMP 2000:05
  • 2. J. Dixmier, Représentations irréductibles des algèbres de Lie nilpotents, An. Acad. Brasil. Ciênc. 35 (1963), 491-519. MR 32:165
  • 3. M. Duflo, Certaines algèbres de type fini sont algèbres de Jacobson, J. Algebra 27 (1973), 358-365. MR 49:9018
  • 4. R. S. Irving, Generic flatness and the Nullstellensatz for Ore extensions, Comm. Algebra 7 (1979), 259-277. MR 80b:13005
  • 5. R. S. Irving, Noetherian algebras and the Nullstellensatz, in Séminaire d'Algèbre Paul Dubreil, Proceedings, Paris 1977-1978, Springer-Verlag, Berlin, 1979, pp. 80-87. MR 81c:16018
  • 6. D. Quillen, On the endomorphism ring of a simple module over an enveloping algebra, Proc. Amer. Math. Soc. 21 (1969), 171-172. MR 39:252

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Additional Information

Ronald S. Irving
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195

Keywords: Generic freeness, Nullstellensatz
Received by editor(s): September 9, 1999
Published electronically: October 31, 2000
Communicated by: Ken Goodearl
Article copyright: © Copyright 2000 American Mathematical Society

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