Algebras and Representation Theory

, Volume 18, Issue 4, pp 931–940 | Cite as

Two-Sided Properties of Elements in Exchange Rings

  • Dinesh Khurana
  • T. Y. Lam
  • Pace P. Nielsen


For any element a in an exchange ring R, we show that there is an idempotent \(\,e\in aR\cap R\,a\,\) such that \(\,1-e\in (1-a)\,R\cap R\,(1-a)\). A closely related result is that a ring R is an exchange ring if and only if, for every aR, there exists an idempotent eR a such that 1−e∈(1−a) R. The Main Theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subsumes both of these results. Finally, applications of these results are given to the study of the endomorphism rings of exchange modules.


Two-sided properties Idempotents Exchange elements Suitable elements Exchange rings Suitable rings Endomorphism rings 

AMS Subject Classifications (2010)

16D40 16D50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Second edition, Graduate Texts in Math., vol. 13. Springer-Verlag, Berlin-Heidelberg-New York (1992)Google Scholar
  2. 2.
    Ara, P., Goodearl, K., O’Meara, K.C., Pardo, E.: Separative cancellation for projective modules over exchange rings. Israel J. Math. 105, 105–137 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Camillo, V., Khurana, D., Lam, T.Y., Nicholson, W.K., Zhou, Y.: Continuous modules are clean. J. Algebras 304, 94–111 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fitting, H.: Die theorie der Automorphismenringen Abelscher Gruppen und ihr Analogon bei nicht kommutativen Gruppen. Math. Ann 107, 514–542 (1933)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fuchs, L: On quasi-injective modules. Ann. Scuola Norm. Sup. Pisa 23, 541–546 (1969)MathSciNetGoogle Scholar
  6. 6.
    Goodearl, K.R.: Von Neumann Regular Rings. Krieger Publishing Company, Malabar (1991)Google Scholar
  7. 7.
    Goodearl, K.R., Warfield, R.B.: Algebras over zero-dimensional rings. Math. Ann. 223, 157–168 (1976)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Khurana, D., Gupta, R.N.: Lifting idempotents and projective covers. Kyungpook Math. J. 41, 217–227 (2001)MathSciNetGoogle Scholar
  9. 9.
    Khurana, D., Lam, T.Y., Nielsen, P.P.: Exchange elements in rings, and the equation X AB X=I. Trans. A.M.S., to appearGoogle Scholar
  10. 10.
    Lam, T.Y.: A First Course in Noncommutative Rings. Second Edition, Graduate Texts in Math., vol. 131. Springer-Verlag, Berlin-Heidelberg-New York (2001)Google Scholar
  11. 11.
    von Neumann, J.: Lectures in Continuous Geometry. Planographed notes, Institute for Advanced Study, Edwards Brothers, Ann Arbor (1937)Google Scholar
  12. 12.
    von Neumann, J.: Continuous Geometry. Edited by I. Halperin, Princeton Univ. Press: Princeton, N.J. (1960)Google Scholar
  13. 13.
    Nicholson, W.K.: Lifting idempotents and exchange rings. Trans. A.M.S. 229, 269–278 (1977)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nicholson, W.K.: On exchange rings. Comm. Alg. 25, 1917–1918 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nicholson, W.K.: Strongly clean rings and Fitting’s lemma. Comm. Algebra 27, 3583–3592 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nicholson, W.K., Zhou, Y.: Strong lifting. J. Alg. 285, 795–818 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Warfield, R.B.: Exchange rings and decompositions of modules. Math. Ann. 199, 31–36 (1972)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsPanjab UniversityChandigarhIndia
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsBrigham Young UniversityProvoUSA

Personalised recommendations