Semicentral Reduced Algebras

  • Gary F. Birkenmeier
  • Jin Yong Kim
  • Jae Keol Park
Part of the Trends in Mathematics book series (TM)

Abstract

An idempotent e of an algebra R is left semicentral if Re = eRe. If 0 and 1 are the only left semicentral idempotents in R, then R is called semicentral reduced. Recent results on generalized triangular matrix algebras and semicentral reduced algebras are surveyed. New results are provided for endomorphism algebras of modules and for semicentral reduced algebras. In particular, semicentral reduced rings which are right FPF, right nonsingular, or left perfect are described.

Keywords

Tral 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, Heidelberg, New York, 1974.CrossRefGoogle Scholar
  2. [2]
    G. F. Birkenmeier, Idempotents and completely semiprime ideals, Comm. Algebra 11 (1983), 567–580.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    G. F. Birkenmeier, A generalization of FPF rings, Comm. Algebra 17 (1989), 855–884.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    G. F. Birkenmeier, Decompositions of Baer-like rings, Acta Math. Hung. 59 (1992), 319–326.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    G. F. Birkenmeier, When does a supernilpotent radical essentially split off?, J. Algebra 172 (1995), 49–60.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    G. F. Birkenmeier, H. E. Heatherly, J. Y. Kim, and J. K. Park, Triangular matrix representations, J. Algebra 230 (2000), 558–595.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    G. F. Birkenmeier, J. Y. Kim, and J. K. Park, A couterexample for CS-rings, Glasgow Math. J. 42 (2000), 263–269.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    G. F. Birkenmeier, B. J. Müller, and S. T. Rizvi, Modules in which every fully invariant submodule is essential in a direct summand, Preprint.Google Scholar
  9. [9]
    K. A. Brown, The singular ideals of group rings, Quart. J. Math. Oxford 28(2) (1977), 41–60.MATHCrossRefGoogle Scholar
  10. [10]
    S. U. Chase, A generalization of the ring of triangular matrices, Nagoya Math. J. 18 (1961), 13–25.MathSciNetMATHGoogle Scholar
  11. [11]
    W. E. Clark, Twisted matrix units semigroup algebras, Duke Math. J. 24 (1967), 417–423.CrossRefGoogle Scholar
  12. [12]
    C. Faith, Injective quotient rings of commutative rings, Module Theory, Lecture Notes in Math. 700, Springer-Verlag, Heidelberg, New York (1979), 151–203.Google Scholar
  13. [13]
    C. Faith and S. Page, FPF Ring Theory: Faithful Modules and Generators of Mod-R, London Math. Soc. Lecture Notes Series 88, Cambridge Univ. Press, Cambridge, 1984.MATHGoogle Scholar
  14. [14]
    K. R. Goodearl, Von Neumann Regular Rings (2nd edition), Krieger, Malabar, 1991.MATHGoogle Scholar
  15. [15]
    T. Y. Lam, A First Course in Noncommutative Rings, SpringerVerlag, Heidelberg, New York, 1991.MATHCrossRefGoogle Scholar
  16. [16]
    J. Lawrence, A singular primitive ring, Proc. Amer. Math. Soc. 45 (1974), 59–62.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Gary F. Birkenmeier
    • 1
  • Jin Yong Kim
    • 2
  • Jae Keol Park
    • 3
  1. 1.Department of MathematicsUniversity of Louisiana at LafayetteLafayetteUSA
  2. 2.Department of MathematicsKyung Hee UniversitySuwonKorea
  3. 3.Department of MathematicsPusan National UniversityPusanKorea

Personalised recommendations