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Rings without infinite sets of noncentral orthogonal idempotents

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Abstract

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.

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References

  1. A. N. Abyzov, “Closure of weakly regular modules with respect to direct sums,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 9, 3–5 (2003).

  2. A. N. Abyzov, “Weakly regular modules over semiperfect rings,” Chebyshevskii Sb., 4, No. 1, 4–9 (2003).

    MATH  MathSciNet  Google Scholar 

  3. A. N. Abyzov, “Weakly regular modules,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 3–6 (2004).

  4. G. Baccella, “Exchange property and the natural preorder between simple modules over semi-Artinian rings,” J. Algebra, 253, 133–166 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  5. W. D. Burgess and W. Stephenson, “Pierce sheaves of non-commutative rings,” Commun. Algebra, 4, No. 1, 51–75 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  6. V. Camillo and H.-P. Yu, “Exchange rings, units and idempotents,” Commun. Algebra, 22, No. 12, 4737–4749 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  7. K. R. Goodearl, Von Neumann Regular Rings, Pitman, London (1979).

    MATH  Google Scholar 

  8. C. Faith, Algebra: Rings, Modules, and Categories I, Springer, Berlin (1973).

    MATH  Google Scholar 

  9. C. Faith, Algebra II, Springer, Berlin (1976).

    MATH  Google Scholar 

  10. H. Hamza, “I 0-rings and I 0-modules,” Math. J. Okayama Univ., 40, 91–97 (1998).

    MathSciNet  Google Scholar 

  11. Kh. I. Khakmi, “Strongly regular and weakly regular rings and modules,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 60–65 (1994).

  12. W. K. Nicholson, “I-rings,” Trans. Am. Math. Soc., 207, 361–373 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  13. W. K. Nicholson, “Lifting idempotents and exchange rings,” Trans. Am. Math. Soc., 229, 269–278 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  14. A. A. Tuganbaev, Distributive Modules and Related Topics, Gordon and Breach, Amsterdam (1999).

    MATH  Google Scholar 

  15. A. A. Tuganbaev, Rings Close to Regular, Kluwer Academic, Dordrecht (2002).

    MATH  Google Scholar 

  16. A. A. Tuganbaev, “Semiregular, weakly regular, and π-regular rings,” J. Math. Sci., 109, No. 3, 1509–1588 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  17. A. A. Tuganbaev, “Modules with many direct summands,” Fundam. Prikl. Mat., 12, No. 8, 233–241 (2006).

    Google Scholar 

  18. A. A. Tuganbaev, “Rings over which all modules are semiregular,” Fundam. Prikl. Mat., 13, No. 2, 185–194 (2007).

    Google Scholar 

  19. A. A. Tuganbaev, “Rings over which all modules are I 0-modules,” Fundam. Prikl. Mat., 13, No. 5, 193–200 (2007).

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia (1991).

    MATH  Google Scholar 

  22. T. Wu, “Exchange with primitive factor rings Artinian,” Algebra Colloq., 3, No. 3, 225–230 (1996).

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Russian University of Trade and Economics, Moscow, Russia

    A. A. Tuganbaev

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  1. A. A. Tuganbaev
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Correspondence to A. A. Tuganbaev.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 14, No. 2, pp. 207–221, 2008.

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Tuganbaev, A.A. Rings without infinite sets of noncentral orthogonal idempotents. J Math Sci 162, 730–739 (2009). https://doi.org/10.1007/s10958-009-9656-z

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  • DOI: https://doi.org/10.1007/s10958-009-9656-z

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