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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Simple algebras of Gelfand-Kirillov dimension two


Author: Jason P. Bell
Journal: Proc. Amer. Math. Soc. 137 (2009), 877-883
MSC (2000): Primary 16P90; Secondary 16P40
Published electronically: October 15, 2008
MathSciNet review: 2457426
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Abstract: Let $ k$ be a field. We show that a finitely generated simple Goldie $ k$-algebra of quadratic growth is noetherian and has Krull dimension $ 1$. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if $ A$ is a finitely generated simple domain of quadratic growth, then $ A$ is noetherian and by a result of Stafford every right and left ideal is generated by at most two elements. We conclude by posing questions and giving examples in which we consider what happens when the hypotheses are relaxed.


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Additional Information

Jason P. Bell
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada, V5A 1S6
Email: jpb@math.sfu.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09724-4
PII: S 0002-9939(08)09724-4
Keywords: GK dimension, quadratic growth, simple rings, noetherian rings.
Received by editor(s): December 21, 2007
Received by editor(s) in revised form: February 21, 2008, and March 17, 2008
Published electronically: October 15, 2008
Additional Notes: The author thanks NSERC for its generous support.
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.