Nonsingular Rings with Finite Type Dimension

  • Yiqiang Zhou
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Two modules are said to be orthogonal if they do not have nonzero isomorphic submodules. An atomic module is any nonzero module whose nonzero submodules are not orthogonal. A module is said to have type dimension n if it contains an essential submodule which is a direct sum of n pairwise orthogonal atomic submodules; If such a number n does not exist, we say the type dimension of this module is ∞. In this paper, we provide characterizations and examples of nonsingular rings with finite type dimension. A characterization theorem is proved for nonsingular rings whose nonzero right ideals contain nonzero atomic right ideals. Type dimension formulas are also obtained for polynomial rings, Laurent polynomial rings and formal triangular matrix rings.

Keywords

Hull 

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References

  1. 1.
    A.H. Al-Huzali, S.K. Jain and S.R. López-Permouth, Rings whose cyclics have finite Goldie dimension, J. Alg. 153 (1992), 37–40.MATHCrossRefGoogle Scholar
  2. 2.
    T.J. Cheatham,Finite dimensional torsion free rings, Pacific J. Math. 39 (1971), 113–118.MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Dauns, Module types, Preprint, 1995.Google Scholar
  4. 4.
    P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.MathSciNetMATHGoogle Scholar
  5. 5.
    A.W. Goldie, The structure of prime rings under ascending chain conditions, Proc. London Math. Soc. 8(1958), 589–608.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    K.R. Goodearl, Singular torsion and the splitting properties,Memoirs Amer. Math. Soc. 124(1972).Google Scholar
  7. 7.
    K.R. Goodearl,Ring Theory: Nonsingular Rings and Modules, Marcel Dekker, Inc (1976).MATHGoogle Scholar
  8. 8.
    J.J. Hutchinson, Quotient full linear rings, Proc. Amer. Math. Soc. 28(1971), 375–378.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    R.E. Johnson, Quotient rings of rings with zero singular ideal, Pacific J. Math. 11(1960), 710–717.MATHGoogle Scholar
  10. 10.
    F. Sandomierski, Semisimple maximal quotient rings, Trans. Amer. Math. Soc. 128(1967), 112–120.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    R.C. Shock, Polynomial rings over finite dimensional rings, Pacific J. Math. 42(1972), 251–258.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    M. Teply, Some aspects of Goldie’s torsion theory, Pacific J. Math. 29(1969), 447–459.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    C.L.Walker and E.A.Walker, Quotient categories and rings of quotient, Rocky Mountain J. Math. 2(1972), 513–555.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    R.W. Wilkerson, Finite dimensional group rings, Proc. Amer. Math. Soc. 41(1973), 10–16.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Y. Zhou, Decomposing modules into direct sums of submodules with types, Preprint, 1996.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Yiqiang Zhou
    • 1
  1. 1.Department of Mathematics and StatisticsMemorial University of New-FoundlandSt. John’sCanada

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