Simplicity of noncommutative Dedekind domains

Authors:
K. R. Goodearl and J. T. Stafford

Journal:
Proc. Amer. Math. Soc. **133** (2005), 681-686

MSC (2000):
Primary 16P40, 16E60

Published electronically:
August 24, 2004

MathSciNet review:
2113915

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The following dichotomy is established: A finitely generated, complex Dedekind domain that is *not* commutative is a simple ring. Weaker versions of this dichotomy are proved for Dedekind prime rings and hereditary noetherian prime rings.

**[ER]**David Eisenbud and J. C. Robson,*Hereditary Noetherian prime rings*, J. Algebra**16**(1970), 86–104. MR**0291222****[FS]**Daniel R. Farkas and Lance W. Small,*Algebras which are nearly finite dimensional and their identities*, Israel J. Math.**127**(2002), 245–251. MR**1900701**, 10.1007/BF02784533**[GS]**M. P. Gilchrist and M. K. Smith,*Noncommutative UFDs are often PIDs*, Math. Proc. Cambridge Philos. Soc.**95**(1984), no. 3, 417–419. MR**755829**, 10.1017/S0305004100061727**[Ja]**A. V. Jategaonkar,*Localization in Noetherian rings*, London Mathematical Society Lecture Note Series, vol. 98, Cambridge University Press, Cambridge, 1986. MR**839644****[Le]**Thomas H. Lenagan,*Krull dimension and invertible ideals in Noetherian rings*, Proc. Edinburgh Math. Soc. (2)**20**(1976), no. 2, 81–86. MR**0419520****[MR]**J. C. McConnell and J. C. Robson,*Noncommutative Noetherian rings*, Revised edition, Graduate Studies in Mathematics, vol. 30, American Mathematical Society, Providence, RI, 2001. With the cooperation of L. W. Small. MR**1811901****[Pa]**Donald S. Passman,*The algebraic structure of group rings*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR**470211****[Pi]**Richard S. Pierce,*Associative algebras*, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. Studies in the History of Modern Science, 9. MR**674652****[Rw]**Louis H. Rowen,*Ring theory. Vol. I*, Pure and Applied Mathematics, vol. 127, Academic Press, Inc., Boston, MA, 1988. MR**940245****[SW1]**J. T. Stafford and R. B. Warfield Jr.,*Hereditary orders with infinitely many idempotent ideals*, J. Pure Appl. Algebra**31**(1984), no. 1-3, 217–225. MR**738216**, 10.1016/0022-4049(84)90087-2**[SW2]**J. T. Stafford and R. B. Warfield Jr.,*Constructions of hereditary Noetherian rings and simple rings*, Proc. London Math. Soc. (3)**51**(1985), no. 1, 1–20. MR**788847**, 10.1112/plms/s3-51.1.1**[Va]**P. Vámos,*On the minimal prime ideal of a tensor product of two fields*, Math. Proc. Cambridge Philos. Soc.**84**(1978), no. 1, 25–35. MR**489566**, 10.1017/S0305004100054840**[Ya]**Sleiman Yammine,*Les théorèmes de Cohen-Seidenberg en algèbre non commutative*, Séminaire d’Algèbre Paul Dubreil 31ème année (Paris, 1977–1978), Lecture Notes in Math., vol. 740, Springer, Berlin, 1979, pp. 120–169 (French). MR**563499**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
16P40,
16E60

Retrieve articles in all journals with MSC (2000): 16P40, 16E60

Additional Information

**K. R. Goodearl**

Affiliation:
Department of Mathematics, University of California, Santa Barbara, California 93106-3080

Email:
goodearl@math.ucsb.edu

**J. T. Stafford**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Email:
jts@umich.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07574-4

Keywords:
Dedekind domain,
simple ring,
invertible ideal,
HNP ring

Received by editor(s):
November 6, 2003

Published electronically:
August 24, 2004

Additional Notes:
The research of both authors was partially supported by grants from the National Science Foundation. Some of it was carried out while the authors participated in the Noncommutative Algebra Year (1999-2000) at the Mathematical Sciences Research Institute in Berkeley, and they thank MSRI for its support

Communicated by:
Lance W. Small

Article copyright:
© Copyright 2004
American Mathematical Society