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The exchange property for purely infinite simple rings

Author: Pere Ara
Journal: Proc. Amer. Math. Soc. 132 (2004), 2543-2547
MSC (2000): Primary 16E50, 16D30
Published electronically: March 25, 2004
MathSciNet review: 2054778
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Abstract: It is proven that every purely infinite simple ring is an exchange ring. This result is applied to determine those Leavitt algebras that are exchange rings.

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  • 1. P. Ara, Extensions of exchange rings, J. Algebra 197 (1997), 409-423. MR 98j:16021
  • 2. P. Ara, K. R. Goodearl, K. C. O'Meara, and E. Pardo, Separative cancellation for projective modules over exchange rings, Israel J. Math. 105 (1998), 105-137. MR 99g:16006
  • 3. P. Ara, M. Gómez Lozano, and M. Siles Molina, Local rings of exchange rings, Comm. in Algebra 26(12) (1998), 4191-4205. MR 99h:16036
  • 4. P. Ara, K. R. Goodearl, and E. Pardo, $K_0$ of purely infinite simple regular rings, $K$-Theory 26 (2002), 69-100.
  • 5. P. Ara, G. K. Pedersen, and F. Perera, An infinite analogue of rings with stable rank one, J. Algebra 230 (2000), 608-655. MR 2001g:16018
  • 6. P. Ara, G. K. Pedersen, and F. Perera, Extensions and pullbacks in $QB$-rings, Algebras and Representation Theory, to appear. Available at ArXiv: math.RA/0107104.
  • 7. G. M. Bergman, Coproducts and some universal ring constructions, Trans. Amer. Math. Soc. 200 (1974), 33-88. MR 50:9971
  • 8. L. G. Brown and G. K. Pedersen, $C^{*}$-algebras of real rank zero, J. Funct. Anal. 99 (1991), 131-149. MR 92m:46086
  • 9. P. M. Cohn, Free Rings and Their Relations, Second Edition, London Mathematical Society Monographs 19, Academic Press, London, 1985. MR 87e:16006
  • 10. P. M. Cohn, On $n$-simple rings, preprint.
  • 11. J. Cuntz, $K$-theory for certain $C^*$-algebras, Annals of Math. 113 (1981), 181-197. MR 84c:46058
  • 12. J. L. García and J. J. Simón, Morita equivalence for idempotent rings, J. Pure and Applied Algebra 76 (1991), 39-56. MR 93b:16010
  • 13. K. R. Goodearl and R. B. Warfield, Jr., Algebras over zero-dimensional rings, Math. Annalen 223 (1976), 157-168. MR 54:357
  • 14. E. Kirchberg, The classification of purely infinite $C^*$-algebras using Kasparov's theory, Fields Institute Comm. Series, 2001 (to appear).
  • 15. W. G. Leavitt, Modules without invariant basis number, Proc. Amer. Math. Soc. 8 (1957), 322-328. MR 18:789a
  • 16. W. G. Leavitt, The module type of a ring, Trans. Amer. Math. Soc. 103 (1962), 113-130. MR 24:A2600
  • 17. N. C. Phillips, A classification theorem for nuclear purely infinite simple $C^*$-algebras, Documenta Math. 5 (2000), 49-114. MR 2001d:46086b
  • 18. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. MR 55:12757
  • 19. M. Rørdam and E. Størmer, Classification of nuclear $C^*$-algebras. Entropy in Operator Algebras, Encyclopedia of Mathematical Sciences, Operator Algebras and Non-Commutative Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 2002. MR 2002i:46047
  • 20. R. B. Warfield, Jr., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31-36. MR 48:11218
  • 21. S. Zhang, A property of purely infinite simple $C^*$-algebras, Proc. Amer. Math. Soc. 109 (1990), 717-720. MR 90k:46134

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Additional Information

Pere Ara
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

Keywords: Purely infinite simple ring, exchange ring, Leavitt algebra
Received by editor(s): March 15, 2003
Received by editor(s) in revised form: May 30, 2003
Published electronically: March 25, 2004
Additional Notes: The author was partially supported by MEC-DGESIC, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.
Communicated by: Martin Lorenz
Article copyright: © Copyright 2004 American Mathematical Society

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