Skip to main content
SpringerLink
Log in

Two-Sided Properties of Elements in Exchange Rings

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

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 a∈R, there exists an idempotent e∈R 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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

  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)

    Article  MathSciNet  Google Scholar 

  3. Camillo, V., Khurana, D., Lam, T.Y., Nicholson, W.K., Zhou, Y.: Continuous modules are clean. J. Algebras 304, 94–111 (2006)

    Article  MathSciNet  Google Scholar 

  4. Fitting, H.: Die theorie der Automorphismenringen Abelscher Gruppen und ihr Analogon bei nicht kommutativen Gruppen. Math. Ann 107, 514–542 (1933)

    Article  MathSciNet  Google Scholar 

  5. Fuchs, L: On quasi-injective modules. Ann. Scuola Norm. Sup. Pisa 23, 541–546 (1969)

    MathSciNet  Google Scholar 

  6. Goodearl, K.R.: Von Neumann Regular Rings. Krieger Publishing Company, Malabar (1991)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Khurana, D., Gupta, R.N.: Lifting idempotents and projective covers. Kyungpook Math. J. 41, 217–227 (2001)

    MathSciNet  Google Scholar 

  9. Khurana, D., Lam, T.Y., Nielsen, P.P.: Exchange elements in rings, and the equation X A−B X=I. Trans. A.M.S., to appear

  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)

  11. von Neumann, J.: Lectures in Continuous Geometry. Planographed notes, Institute for Advanced Study, Edwards Brothers, Ann Arbor (1937)

  12. von Neumann, J.: Continuous Geometry. Edited by I. Halperin, Princeton Univ. Press: Princeton, N.J. (1960)

  13. Nicholson, W.K.: Lifting idempotents and exchange rings. Trans. A.M.S. 229, 269–278 (1977)

    Article  MathSciNet  Google Scholar 

  14. Nicholson, W.K.: On exchange rings. Comm. Alg. 25, 1917–1918 (1997)

    Article  MathSciNet  Google Scholar 

  15. Nicholson, W.K.: Strongly clean rings and Fitting’s lemma. Comm. Algebra 27, 3583–3592 (1999)

    Article  MathSciNet  Google Scholar 

  16. Nicholson, W.K., Zhou, Y.: Strong lifting. J. Alg. 285, 795–818 (2005)

    Article  MathSciNet  Google Scholar 

  17. Warfield, R.B.: Exchange rings and decompositions of modules. Math. Ann. 199, 31–36 (1972)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Panjab University, Chandigarh, 160014, India

    Dinesh Khurana

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    T. Y. Lam

  3. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Pace P. Nielsen

Authors
  1. Dinesh Khurana
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Y. Lam
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pace P. Nielsen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pace P. Nielsen.

Additional information

Presented by Kenneth Goodearl.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khurana, D., Lam, T.Y. & Nielsen, P.P. Two-Sided Properties of Elements in Exchange Rings. Algebr Represent Theor 18, 931–940 (2015). https://doi.org/10.1007/s10468-015-9524-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-015-9524-0

Keywords

AMS Subject Classifications (2010)

Search

Navigation