Skip to main content

Stability Properties of Exchange Rings

  • Conference paper
International Symposium on Ring Theory

Part of the book series: Trends in Mathematics ((TM))

Abstract

We survey some recent results on exchange rings,with emphasis on stability theorems in K-theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Ara, Aleph-nought-continuous regular rings, J. Algebra 109 (1987), 115–126.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Ara, Left and right projections are equivalent in Rickart C * - algebras, J. Algebra 120 (1989), 433–448.

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Ara, Strongly π-regular rings have stable range one, Proc. Amer. Math. Soc. 124 (1996), 2293–2298.

    Article  MathSciNet  Google Scholar 

  4. P. Ara, Extensions of exchange rings, J. Algebra 197 (1997), 409–423.

    Article  MathSciNet  MATH  Google Scholar 

  5. 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.

    MathSciNet  MATH  Google Scholar 

  6. P. Ara, K. R. Goodearl, K. C. O’Meara and R. Raphael, K 1 of separative exchange rings and C * -algebras with real rank zero, Pacific J. Math. 195 (2000), 261–275.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Ara, K. R. Goodearl, E. Pardo, and D. V. Tyukavkin, K-theoretically simple von Neumann regular rings, J. Algebra 174 (1995), 659–677.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Ara, M. Gómez Lozano and M. Siles Molina, Local rings of exchange rings, Comm Algebra 26(12) (1998), 4191–4205.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Ara, K. C. O’Meara and D. V. Tyukavkin, Cancellation of projective modules over regular rings with comparability, J. Pure and Applied Algebra 107 (1996), 19–38.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Ara and E. Pardo, Refinement monoids with weak comparability and applications to regular rings and C * -algebras, Proc. Amer. Math. Soc. 124 (1996), 715–720.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Ara, E. Pardo and F. Perera, The structure of countably generated projective modules over regular rings, J. Algebra 226 (2000), 161–190.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Ara and F. Perera, Multipliers of von Neumann regular rings, Comm Algebra 28 (2000), 3359–3385.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Baccella, Right semiartinian rings are exchange rings,Preprint.

    Google Scholar 

  14. G. Brookfield, Direct sum cancellation of Noetherian modules, J. Algebra 200 (1998), 207–224.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. G. Brown and G. K. Pedersen, C * -algebras of real rank zero, J. Functional Analysis 99 (1991), 131–149.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. G. Brown and G. K. Pedersen, On the geometry of the unit ball of a C * -algebra, J. Reine Angew. Math. 469 (1995), 113–147.

    MathSciNet  MATH  Google Scholar 

  17. L. G. Brown and G. K. Pedersen, Non-stable K -theory and extremally rich C * -algebras,in preparation.

    Google Scholar 

  18. H. Chen and F.-U. Li,Whitehead groups of exchange rings with primitive factors artinian,preprint.

    Google Scholar 

  19. A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol 1, Math. Surveys 7, Amer. Math. Soc., Providence, 1961.

    MATH  Google Scholar 

  20. P. Crawley and B. Jónsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797–855.

    Article  MathSciNet  MATH  Google Scholar 

  21. E. G. Evans, Jr., Krull-Schmidt and cancellation over local rings, Pacific J. Math. 46 (1973), 115–121.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Facchini, Module Theory:Endomorphism rings and direct sum decompositions in some classes of modules, Progress in Math. 167, Birkhäuser, Basel, 1998.

    MATH  Google Scholar 

  23. C. Faith and Y. Utumi, On a new proof of Litoff ‘s theorem, Acta Math. Acad. Sci. Hungar. 14 (1963), 369–371.

    Article  MathSciNet  MATH  Google Scholar 

  24. K. R. Goodearl, Direct sum properties of quasi-injective modules, Bull. Amer. Math. Soc. 82 (1976), 108–110.

    Article  MathSciNet  MATH  Google Scholar 

  25. K. R. Goodearl, Von Neumann Regular Rings, Pitman, London, 1979, Second Ed., Krieger, Malabar, 1991.

    Google Scholar 

  26. K. R. Goodearl, Partially Ordered Abelian Groups with Interpolation, Math. Surveys and Monographs, 20, Amer. Math. Soc., Providence, 1986.

    MATH  Google Scholar 

  27. K. R. Goodearl and R. B. Warfield, Jr., Algebras over zero- dimensional rings, Math. Ann 223 (1976), 157–168.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Harada, Factor Categories with Applications to Direct De- compositions of Modules, Marcel Dekker, New York, 1983.

    Google Scholar 

  29. M. Harada and T. Ishii, On perfect rings and the exchange property, Osaka J. Math. 12 (1975), 483–491.

    MathSciNet  MATH  Google Scholar 

  30. T.-Y. Lam, Modules with isomorphic multiples and rings with isomorphic matrix rings, a survey, Monographie 35 de l’Enseignement Mathématique, Genève, 1999.

    MATH  Google Scholar 

  31. P. Menal and J. Moncasi, Lifting units in self-injective rings and an index theory for Rickart C *-algebras, Pacific J. Math. 126 (1987), 295–329.

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Menal and J. Moncasi, K 1 of von Neumann regular rings, J. Pure and Applied Algebra 33 (1984), 295–312.

    Article  MathSciNet  MATH  Google Scholar 

  33. S. H. Mohamed and B. J. Mueller, Continuous and discrete modules, London Math. Soc. Lecture Note Series 147 Cambridge Univ. Press, Cambridge, 1990.

    Book  MATH  Google Scholar 

  34. S. H. Mohamed and B. J. Mueller, On the exchange property for quasi-continuous modules, in Abelian groups and modules (Padova, 1994), 367–372, Math. Appl. 343, Kluwer Acad. Publ., Dordrecht, 1995.

    Google Scholar 

  35. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269–278.

    Article  MathSciNet  MATH  Google Scholar 

  36. W. K. Nicholson, On exchange rings, Comm in Algebra 25(6) (1997), 1917–1918.

    Article  MathSciNet  MATH  Google Scholar 

  37. K. C. O’Meara, Simple regular rings satisfying weak comparability, J. Algebra 141 (1991), 162–186.

    Article  MathSciNet  MATH  Google Scholar 

  38. K. C. O’Meara, The exchange property for row and column-finite matrix rings,J. Algebra,to appear.

    Google Scholar 

  39. K. C. O’Meara and C. Vinsonhaler, Separative cancellation and multi-isomorphism in torsion-free abelian groups, J. Algebra 221 (1999), 536–550.

    Article  MathSciNet  MATH  Google Scholar 

  40. K. Oshiro, Projective modules over von Neumann regular rings have the finite exchange property, Osaka J.Math. 20 (1983), 695–699.

    MathSciNet  MATH  Google Scholar 

  41. K. Oshiro and S. T. Rizvi, The exchange property of quasi-continuous modules with the finite exchange property, Osaka J. Math. 33 (1996), 217–234.

    MathSciNet  MATH  Google Scholar 

  42. E. Pardo, Monoides de refinament i anells d’intercanvi, Ph.D. Thesis, Universitat Autonoma de Barcelona, 1995.

    Google Scholar 

  43. E. Pardo, Comparability, separativity, and exchange rings, Comm Algebra 24(9) (1996), 2915–2929.

    Article  MathSciNet  MATH  Google Scholar 

  44. F. Perera, The structure of positive elements for C * -algebras with real rank zero, International J. Math. 8 (1997), 383–405.

    Article  MathSciNet  MATH  Google Scholar 

  45. F. Perera, Ideal structure of multiplier algebras of simple C * -algebras with real rank zero,Canad. J. Math.,to appear.

    Google Scholar 

  46. F. Perera, Lifting units modulo exchange ideals and C * -algebras with real rank zero, J. Reine Angew. Math. 522 (2000), 51–62.

    MathSciNet  MATH  Google Scholar 

  47. J. Rosenberg, Algebraic K-Theory and Its Applications, Grad. Texts in Math. 147, Springer-Verlag, Heidelberg, New York, 1994.

    Book  MATH  Google Scholar 

  48. J. Stock, On rings whose projective modules have the exchange property, J. Algebra 103 (1986), 437–453.

    Article  MathSciNet  MATH  Google Scholar 

  49. R. B. Warfield, Jr., Exchange rings and decompositions of modules, Math. Aim 199 (1972), 31–36.

    MathSciNet  MATH  Google Scholar 

  50. R. B. Warfield, Jr., Cancellation of modules and groups and stable range of endomorphism rings, Pacific J. Math. 91 (1980), 457–485.

    Article  MathSciNet  MATH  Google Scholar 

  51. F. Wehrung, Injective positively ordered monoids I, J. Pure and Applied Algebra 83 (1992), 43–82.

    Article  MathSciNet  MATH  Google Scholar 

  52. Tongsuo Wu and Yonghua Xu, On the stable range condition of exchange rings,Comm Algebra 25(7) (1997),2355–2363.

    Article  MathSciNet  MATH  Google Scholar 

  53. K. Yamagata, On rings of finite representation type and modules with the finite exchange property, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 13 (1975), 1–6.

    MathSciNet  MATH  Google Scholar 

  54. H.-P. Yu, Stable range one for exchange rings, J. Pure and Applied Algebra 98 (1995), 105–109.

    MathSciNet  MATH  Google Scholar 

  55. B. Zimmermann-Huisgen, Exchanging torsion modules over Dede-kind domains, Arch. Math. 55 (1990), 241–246.

    Article  MathSciNet  MATH  Google Scholar 

  56. B. Zimmermann-Huisgen and W. Zimmermann, Classes of modules with the exchange property, J. Algebra 88 (1984), 416–434.

    Article  MathSciNet  MATH  Google Scholar 

  57. S. Zhang,Certain C * -algebras with real rank zero and their corona and multiplier algebras, Part I, Pacific J. Math. 155 (1992), 169–197.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Science+Business Media New York

About this paper

Cite this paper

Ara, P. (2001). Stability Properties of Exchange Rings. In: Birkenmeier, G.F., Park, J.K., Park, Y.S. (eds) International Symposium on Ring Theory. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0181-6_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0181-6_2

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6650-1

  • Online ISBN: 978-1-4612-0181-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics