Rings with finite decomposition of identity
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  • 作者：M. A. Dokuchaev (1)
    N. M. Gubareni (2)
    V.V. Kirichenko (3)
  • 刊名：Ukrainian Mathematical Journal
  • 出版年：2011
  • 出版时间：August 2011
  • 年：2011
  • 卷：63
  • 期：3
  • 页码：369-392
  • 全文大小：259.2 KB
  • 参考文献：1. R. Baer, “Nets and groups,” Trans. Amer. Math. Soc., 46, 110–141 (1939).
    2. R. Baer, “Nets and groups. II,” Trans. Amer. Math. Soc., 47, 435–439 (1940).
    3. H. Bass, “Finitistic dimension and homological generalization of semiprimary rings,” Trans. Amer. Math. Soc., 95, 466–488 (1960).
    4. R. Bautista, “On algebras close to hereditary Artin algebras,” An. Inst. Mat. Univ. Nac. Aut贸n. M茅x., 20, No. 1, 21–104 (1981).
    5. R. Bautista and D. Simson, “Torsion modules over l-Gorenstein l-hereditary Artinian rings,” Commun. Algebra, 12, No. 7–8, 899–936 (1984).
    6. G. Birkhoff, “Lattice theory,” Amer. Math. Soc. Colloq. Publ., Amer. Math. Soc., New York, Vol. 25 (1948).
    7. Z. I. Borevich, “Parabolic subgroups in a special linear group over a semilocal ring,” Vestn. Leningrad. Univ., Mat., Mekh., Astronom., 4, No. 19, S. 29–34 (1976).
    8. Z. I. Borevich and B. A. Tolasov, “Net normal subgroups of the general linear group,” Rings and modules: Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 64, 49–54, 159–160 (1976).
    9. Z. I. Borevich and E. V. Dybkova, “The index of net subgroups in general and special linear groups over a Dedekind ring,” Algebra, Number Theory, and Their Appl.: Trudy Mat. Inst. Steklov., 148, 58–64 (1978).
    10. Z. I. Borevich and Kh. O. Lesama Serrano, “Multiplicative group of a semiperfect ring,” Visn. Kiev. Mat., Mech., 27, 19–20 (1985).
    11. V. Camillo and H.-P. Yu, “Exchange rings, units, and idempotents,” Commun. Algebra, 22(12), 4737–4749 (1994).
    12. S. U. Chase, “A generalization of the ring of triangular matrix,” Nagoya Math. J., 18, 13–25 (1961).
    13. P. M. Cohn, “B茅zout rings and their subrings,” Proc. Cambridge Phil. Soc., 64, 251–264 (1968).
    14. P. Crawley and B. J贸nsson, “Refinements for infinite direct decompositions of algebraic systems,” Pacif. J. Math., 14, 797–855 (1964).
    15. M. A. Dokuchaev, N. M. Gubareni, and V. V. Kirichenko, “Semiperfect ipri-rings and right B茅zout rings,” Ukr. Mat. Zh., 62, No. 5, 612–624 (2010); English translation: Ukr. Math. J., 62, No. 5, 701–715 (2010).
    16. M. A. Dokuchaev, V. V. Kirichenko, B. V. Novikov, and A. P. Petravchuk, “On incidence modulo ideal rings,” J. Algebra Appl., 6, No. 4, 553–586 (2007).
    17. Yu. A. Drozd, “The structure of hereditary rings,” Mat. Sb., 113(155) (1980); English translation: Math. Sbornik (USSR), 41, No. 1, 139–148 (1982).
    18. E. V. Dybkova, “On net subgroups in a hyperbolic unitary group,” Algebra Anal., 9, No. 4, 79–86 (1997); English translation: St.Petersburg Math. J., 9, No. 4, 725–731 (1998).
    19. K. R. Fuller, “Rings of left invariant module type,” Commun. Algebra, 6, 153–167 (1978).
    20. K. R. Goodearl, Von Neumann Regular Rings, 2nd ed., Krieger Publ. Co., Malabar, Florida (1991).
    21. A. W. Goldie, “Semi-prime rings with maximum condition,” Proc. London Math. Soc., 10, No. 3, 201–210 (1960).
    22. R. Gordon and L. W. Small, “Piecewise domains,” J. Algebra, 23, 553–564 (1972).
    23. H. Harada, “Hereditary semi-primary rings and triangular matrix rings,” Nagoya Math. J., 27, 463–484 (1966).
    24. M. Hazewinkel, N. Gubareni, and V. V. Kirichenko, Algebras, Rings, and Modules, Vol. 1, Kluwer, Dordrecht (2004).
    25. N. Jacobson, “Some remarks on one-sided inverses,” Proc. Amer. Math. Soc., 1, 352–355 (1950).
    26. C. U. Jensen, “On characterizations of Pr篓ufer rings,” Math. Scand., 13, 90–98 (1963).
    27. I. Kaplansky, “Elementary divisors and modules,” Trans. Amer. Math. Soc., 66, 464–491 (1949).
    28. V. V. Kirichenko and M. A. Khibina, “Semiperfect semidistributive rings”, in: Infinite Groups and Related Algebraic Structures [in Russian], Kiev (1993), pp. 457–480.
    29. T. Y. Lam, “First course in noncommutative rings,” Grad. Texts Math., 131 (1991).
    30. T. Y. Lam, Lectures on Modules and Rings, Springer (1999).
    31. B. J. Muller and S. Singh, “Uniform modules over serial rings,” J. Algebra, 144, No. 1, 94–109 (1991).
    32. W. K. Nicholson, “I-rings,” Trans. Amer. Math. Soc., 207, 361–371 (1975).
    33. L. H. Rowen, Ring Theory, Vol. 1 Academic Press, New York (1988).
    34. P. L. Sandomerski, “A note on the global dimension of subrings,” Proc. Amer. Math. Soc., 23, 478–480 (1969).
    35. M. Shirvani and B. A. F. Wehrfritz, Skew Linear Groups, Cambridge Univ. Press (1986).
    36. L. A. Skornyakov, “On Cohn rings,” Algebra Logika, 4, No. 3, 5–30 (1965).
    37. L. W. Small, “Semihereditary rings,” Bull. Amer. Math. Soc., 73, 656–659 (1967).
    38. L. W. Small, “Hereditary rings,” Proc. Nat. Acad. Soc., 55, 25–27 (1966).
    39. J. C. Shepherdson, “Inverses and zero divisors in matrix rings,” Proc. London Math. Soc., 1, 71–85 (1951).
    40. A. A. Tuganbaev, “Distributive rings and modules,” Trudy Mosk. Mat. Obshch., 51, 95–113 (1988).
    41. A. A. Tuganbaev, Semidistributive Modules and Rings, Kluwer, Dordrecht (1998).
    42. A. A. Tuganbaev, “Modules with the exchange properties and exchange rings,” in: M. Hazewinkel (editor), Handbook of Algebra, Vol. 2, North-Holland, Amsterdam (2000), pp. 439–459.
    43. N. A. Vavilov and E. B. Plotkin, “Net subgroups of Chevalley groups. II. Gaussian decomposition,” Modules and Algebraic Groups: Zap. Nauchn. Sem. Leningrad. Otdelen. Mat. Inst. Steklov (LOMI), 114, 62–76, 218–219 (1982).
    44. N. A. Vavilov and E. B. Plotkin, “Net subgroups of Chevalley groups,” Rings and Modules, 2: Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 94, 40–49, 150 (1979).
    45. R. B. Warfield, Jr., “A Krull–Schmidt theorem for infinite sums of modules,” Proc. Amer. Math. Soc., 22, 460–465 (1969).
    46. R. B. Warfield, Jr., “Exchange rings and decompositions of modules,” Math. Ann., 199, 31–36 (1972).
    47. R. B. Warfield, Jr., “Serial rings and finitely presented modules,” J. Algebra, 37, No. 2, 187–222 (1975).
    48. A. E. Zalesski, “Sylow p-subgroups of the general linear group over a skew field,” Izv. Akad. Nauk SSSR, Ser. Mat., 31, 1149–1958 (1967).
  • 作者单位：1. University of S茫o Paulo, S茫o Paulo, Brazil2. Cze蜁stochowa University of Technology, Cze蜁stochowa, Poland3. Shevchenko Kiev National University, Kiev, Ukraine
  • 刊物类别：Mathematics and Statistics
  • 刊物主题：Mathematics
    Mathematics
    Analysis
    Applications of Mathematics
    Geometry
    Statistics
    Algebra
    Russian Library of Science
  • 出版者：Springer New York
  • ISSN：1573-9376
文摘
A criterion for semiprime rings with finite decomposition of identity to be prime is given. We also present a brief survey of some finiteness conditions related to the decomposition of identity. We consider the notion of a net of a ring and show that the lattice of all two-sided ideals of a right semidistributive semiperfect ring is distributive. An application of decompositions of identity to groups of units is given.

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