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Primitive noetherian algebras with big centers
Author(s):
Ronald
S.
Irving
Journal:
Proc. Amer. Math. Soc.
129
(2001),
1587-1593.
MSC (2000):
Primary 16D60, 16P40;
Secondary 16S36
Posted:
October 31, 2000
MathSciNet review:
1814084
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Abstract:
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:
- 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
Email:
irving@math.washington.edu
DOI:
10.1090/S0002-9939-00-05809-3
PII:
S 0002-9939(00)05809-3
Keywords:
Generic freeness,
Nullstellensatz
Received by editor(s):
September 9, 1999
Posted:
October 31, 2000
Communicated by:
Ken Goodearl
Copyright of article:
Copyright
2000,
American Mathematical Society
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