Rings without infinite sets of noncentral orthogonal idempotents



Let A be a ring without infinite sets of noncentral orthogonal idempotents. A is an exchange ring if and only if all Pierce stalks of A are semiperfect rings. All A-modules are I 0-modules if and only if either A is a right semi-Artinian ring in which every proper right ideal is the intersection of maximal right ideals or A/ SI(A A ) is an Artinian serial ring such that the square of the Jacobson radical of A/ SI(A A ) is equal to zero.


Exchange Ring Special Ring Central Idempotent Factor Ring Essential Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    A. N. Abyzov, “Closure of weakly regular modules with respect to direct sums,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 9, 3–5 (2003).Google Scholar
  2. 2.
    A. N. Abyzov, “Weakly regular modules over semiperfect rings,” Chebyshevskii Sb., 4, No. 1, 4–9 (2003).MATHMathSciNetGoogle Scholar
  3. 3.
    A. N. Abyzov, “Weakly regular modules,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 3–6 (2004).Google Scholar
  4. 4.
    G. Baccella, “Exchange property and the natural preorder between simple modules over semi-Artinian rings,” J. Algebra, 253, 133–166 (2002).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    W. D. Burgess and W. Stephenson, “Pierce sheaves of non-commutative rings,” Commun. Algebra, 4, No. 1, 51–75 (1976).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    V. Camillo and H.-P. Yu, “Exchange rings, units and idempotents,” Commun. Algebra, 22, No. 12, 4737–4749 (1994).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    K. R. Goodearl, Von Neumann Regular Rings, Pitman, London (1979).MATHGoogle Scholar
  8. 8.
    C. Faith, Algebra: Rings, Modules, and Categories I, Springer, Berlin (1973).MATHGoogle Scholar
  9. 9.
    C. Faith, Algebra II, Springer, Berlin (1976).MATHGoogle Scholar
  10. 10.
    H. Hamza, “I 0-rings and I 0-modules,” Math. J. Okayama Univ., 40, 91–97 (1998).MathSciNetGoogle Scholar
  11. 11.
    Kh. I. Khakmi, “Strongly regular and weakly regular rings and modules,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 60–65 (1994).Google Scholar
  12. 12.
    W. K. Nicholson, “I-rings,” Trans. Am. Math. Soc., 207, 361–373 (1975).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    W. K. Nicholson, “Lifting idempotents and exchange rings,” Trans. Am. Math. Soc., 229, 269–278 (1977).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A. A. Tuganbaev, Distributive Modules and Related Topics, Gordon and Breach, Amsterdam (1999).MATHGoogle Scholar
  15. 15.
    A. A. Tuganbaev, Rings Close to Regular, Kluwer Academic, Dordrecht (2002).MATHGoogle Scholar
  16. 16.
    A. A. Tuganbaev, “Semiregular, weakly regular, and π-regular rings,” J. Math. Sci., 109, No. 3, 1509–1588 (2002).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    A. A. Tuganbaev, “Modules with many direct summands,” Fundam. Prikl. Mat., 12, No. 8, 233–241 (2006).Google Scholar
  18. 18.
    A. A. Tuganbaev, “Rings over which all modules are semiregular,” Fundam. Prikl. Mat., 13, No. 2, 185–194 (2007).Google Scholar
  19. 19.
    A. A. Tuganbaev, “Rings over which all modules are I 0-modules,” Fundam. Prikl. Mat., 13, No. 5, 193–200 (2007).MathSciNetGoogle Scholar
  20. 20.
    R. B. Warfield, “Exchange rings and decompositions of modules,” Math. Ann., 199, 31–36 (1972).MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia (1991).MATHGoogle Scholar
  22. 22.
    T. Wu, “Exchange with primitive factor rings Artinian,” Algebra Colloq., 3, No. 3, 225–230 (1996).MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Russian University of Trade and EconomicsMoscowRussia

Personalised recommendations