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Rendiconti del Circolo Matematico di Palermo

, Volume 35, Issue 3, pp 349–363 | Cite as

On rings whose number of centralizers of ideals is finite

  • K. Kishimoto
  • J. Krempa
  • A. Nowicki
Article

Abstract

LetR be a ring. For the setF of all nonzero ideals ofR, we introduce an equivalence relation inF as follows: For idealsI andJ, I∼J if and only ifV R (I)=V R(J), whereV R() is the centralizer inR. LetI R=F/∼. Then we can see thatn(I R), the cardinality ofI R, is 1 if and only ifR is either a prime ring or a commutative ring (Theorem 1.1). An idealI ofR is said to be a commutator ideal ifI is generated by{st−ts; s∈S, t∈T} for subsetS andT ofR, andR is said to be a ring with (N) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem: LetR be a ring with (N). Thenn(I R) is finite if and only ifR is isomorphic to an irredundant subdirect sum ofS⊕Z whereS is a finite direct sum of non commutative prime rings andZ is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ringR such thatn(I R)=m for any given natural numberm.

Keywords

Prime Ideal Commutative Ring Prime Ring Nonzero Ideal Maximal Class 
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]
    Herstein I.N.,Rings with involution, Chicago Lecture in Mathematics, The University of Chicago Press, 1976.Google Scholar
  2. [2]
    Jacobson N.,The theory of rings, Mathematical Survey, No. II, A.M.S. Providence, 1943.MATHGoogle Scholar
  3. [3]
    Nowicki A.,Derivations of special subrings of matrix rings and regular graphs, Tsukuba J. Math.7 (1983), 381–297.MathSciNetGoogle Scholar

Copyright information

© Springer 1986

Authors and Affiliations

  • K. Kishimoto
    • 1
    • 2
    • 3
  • J. Krempa
    • 1
    • 2
    • 3
  • A. Nowicki
    • 1
    • 2
    • 3
  1. 1.Department of MathematicsShinshu UniversityN. MatsumotoJapan
  2. 2.Institute of MathematicsUniversity of WarsawWarsawPoland
  3. 3.Institute of MathematicsN. Copernicus UniversityTorunPoland

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