The exchange property for purely infinite simple rings

Ara, Pere

doi:10.1090/S0002-9939-04-07369-1

The exchange property for purely infinite simple rings
HTML articles powered by AMS MathViewer

by Pere Ara PDF

Proc. Amer. Math. Soc. 132 (2004), 2543-2547 Request permission

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.

References

Pere Ara, Extensions of exchange rings, J. Algebra 197 (1997), no. 2, 409–423. MR 1483771, DOI 10.1006/jabr.1997.7116
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 1639739, DOI 10.1007/BF02780325
Pere Ara, Miguel Gómez Lozano, and Mercedes Siles Molina, Local rings of exchange rings, Comm. Algebra 26 (1998), no. 12, 4191–4205. MR 1661292, DOI 10.1080/00927879808826405
P. Ara, K. R. Goodearl, and E. Pardo, $K_0$ of purely infinite simple regular rings, $K$-Theory 26 (2002), 69–100.
Pere Ara, Gert K. Pedersen, and Francesc Perera, An infinite analogue of rings with stable rank one, J. Algebra 230 (2000), no. 2, 608–655. MR 1775806, DOI 10.1006/jabr.2000.8330
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.
George M. Bergman, Coproducts and some universal ring constructions, Trans. Amer. Math. Soc. 200 (1974), 33–88. MR 357503, DOI 10.1090/S0002-9947-1974-0357503-7
Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI 10.1016/0022-1236(91)90056-B
P. M. Cohn, Free rings and their relations, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR 800091
P. M. Cohn, On $n$-simple rings, preprint.
Joachim Cuntz, $K$-theory for certain $C^{\ast }$-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197. MR 604046, DOI 10.2307/1971137
José Luis García and Juan Jacobo Simón, Morita equivalence for idempotent rings, J. Pure Appl. Algebra 76 (1991), no. 1, 39–56. MR 1140639, DOI 10.1016/0022-4049(91)90096-K
K. R. Goodearl and R. B. Warfield Jr., Algebras over zero-dimensional rings, Math. Ann. 223 (1976), no. 2, 157–168. MR 412230, DOI 10.1007/BF01360879
E. Kirchberg, The classification of purely infinite $C^*$-algebras using Kasparov’s theory, Fields Institute Comm. Series, 2001 (to appear).
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
W. G. Leavitt, The module type of a ring, Trans. Amer. Math. Soc. 103 (1962), 113–130. MR 132764, DOI 10.1090/S0002-9947-1962-0132764-X
Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact $C^*$-algebras in the Cuntz algebra $\scr O_2$, J. Reine Angew. Math. 525 (2000), 17–53. MR 1780426, DOI 10.1515/crll.2000.065
W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269–278. MR 439876, DOI 10.1090/S0002-9947-1977-0439876-2
M. Rørdam and E. Størmer, Classification of nuclear $C^*$-algebras. Entropy in operator algebras, Encyclopaedia of Mathematical Sciences, vol. 126, Springer-Verlag, Berlin, 2002. Operator Algebras and Non-commutative Geometry, 7. MR 1878881, DOI 10.1007/978-3-662-04825-2
R. B. Warfield Jr., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31–36. MR 332893, DOI 10.1007/BF01419573
Shuang Zhang, A property of purely infinite simple $C^*$-algebras, Proc. Amer. Math. Soc. 109 (1990), no. 3, 717–720. MR 1010004, DOI 10.1090/S0002-9939-1990-1010004-X