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Ring Hulls of Semiprime Homomorphic Images

  • Gary F. Birkenmeier
  • Jae Keol Park
  • S. Tariq Rizvi
Part of the Trends in Mathematics book series (TM)

Abstract

We investigate connections between the right FI-extending right ring hulls of semiprime homomorphic images of a ring R and the right FI-extending right rings of quotients of R by considering ideals of R which are essentially closed and contain the prime radical P(R). As an application of our results, we show that the bounded central closure of a unital C*-algebra A contains a nonzero homomorphic image of A/K for every nonessential ideal K of A.

Keywords

Homomorphic Image Semiprime Ring Proper Ideal Injective Hull Central Closure 
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.

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References

  1. [1]
    P. Ara and M. Mathieu, Local Multipliers of C*-Algebras, Springer Monographs in Math., Springer-Verlag, London, 2003Google Scholar
  2. [2]
    K. Beidar and R. Wisbauer, Strongly and properly semiprime modules and rings, Ring Theory, Proc. Ohio State-Denison Conf. (S.K. Jain and S.T. Rizvi (eds.)), World Scientific, Singapore (1993), 58–94Google Scholar
  3. [3]
    G.F. Birkenmeier, Idempotents and completely semiprime ideals, Comm. Algebra 11 (1983), 567–580MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    G.F. Birkenmeier, Decompositions of Baer-like rings, Acta Math. Hungar. 59 (1992), 319–326MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    G.F. Birkenmeier, When does a supernilpotent radical split off?, J. Algebra 172 (1995), 49–60MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    G.F. Birkenmeier, G. CĂlugĂreanu, L. Fuchs and H.P. Goeters, The fully-invariant-extending property for Abelian groups, Comm. Algebra 29 (2001), 673–685MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    G.F. Birkenmeier, H.E. Heatherly, J.Y. Kim and J.K. Park, Triangular matrix representations, J. Algebra 230 (2000), 558–595MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    G.F. Birkenmeier, J.Y. Kim and J.K. Park, Quasi-Baer ring extensions and biregular rings, Austral. Math. Soc. 61 (2000), 39–52MATHMathSciNetGoogle Scholar
  9. [9]
    G.F. Birkenmeier, B.J. Müller and S.T. Rizvi, Modules in which every fully invariant submodule is essential in a direct summand, Comm. Algebra 30 (2002), 1395–1415MATHMathSciNetGoogle Scholar
  10. [10]
    G.F. Birkenmeier, J.K. Park and S.T. Rizvi, Modules with fully invariant submodules essential in fully invariant summands, Comm. Algebra 30 (2002), 1833–1852MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    G.F. Birkenmeier, J.K. Park and S.T. Rizvi, Ring hulls and applications, J. Algebra 304 (2006), 633–665MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    G.F. Birkenmeier, J.K. Park and S.T. Rizvi, Hulls of semiprime rings with applications to C*-algebras, PreprintGoogle Scholar
  13. [13]
    G.F. Birkenmeier, J.K. Park and S.T. Rizvi, The structure of rings of quotients, PreprintGoogle Scholar
  14. [14]
    W.E. Clark, Twisted matrix units semigroup algebras, Duke Math. J. 34 (1967), 417–424MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    J. Dixmier, C*-Algebras, North-Holland (1977)Google Scholar
  16. [16]
    G.A. Elliott, Automorphisms determined by multipliers on ideals of a C*-algebra, J. Funct. Anal. 23 (1976), 1–10MATHCrossRefGoogle Scholar
  17. [17]
    C. Faith and Y. Utumi, Maximal quotient rings, Proc. Amer. Math. Soc. 16 (1965), 1084–1089MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    K.R. Goodearl, Ring Theory: Nonsingular Rings and Modules, Marcel Dekker, New York (1976)MATHGoogle Scholar
  19. [19]
    J. Lambek, Lectures on Rings and Modules, Chelsea, New York (1986)Google Scholar
  20. [20]
    A.C. Mewborn, Regular rings and Baer rings, Math. Z. 121 (1971), 211–219MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    G.K. Pedersen, Approximating derivations on ideals of C*-algebras, Invent. Math. 45 (1978), 299–305MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    J.C. Robson and L.W. Small, Liberal extensions, Proc. London Math. Soc. (3)42 (1981), 83–103MathSciNetGoogle Scholar
  23. [23]
    C.L. Walker and E.A. Walker, Quotient categories and rings of quotients, Rocky Mountain J. Math. 2 (1972), 513–555MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Gary F. Birkenmeier
    • 1
  • Jae Keol Park
    • 2
  • S. Tariq Rizvi
    • 3
  1. 1.Department of MathematicsUniversity of Louisiana at LafayetteLafayetteUSA
  2. 2.Department of MathematicsBusan National UniversityBusanSouth Korea
  3. 3.Department of MathematicsOhio State UniversityLimaUSA

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