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Distributive rings with maximum conditions

  • Askar A. Tuganbaev
Part of the Mathematics and Its Applications book series (MAIA, volume 449)

Abstract

Let N be a completely prime ideal of a right distributive ring A, and let \( N \subseteq J(A). \) Then the following assertions hold.
  1. (1)

    N = mN ⊂ mA for all m ∈ A\N, and A has no nontrivial idempotents.

     
  2. (2)

    N is comparable to any right ideal of A.

     
  3. (3)

    M N = N for any right ideal M of A which is not contained in N.

     
  4. (4)

    Either N = 0 and A is a right uniform domain, or N is a nonzero essential right ideal of A.

     
  5. (5)

    Either the module NA IS not uniform, or A is right uniform.

     
  6. (6)

    If N is a finitely generated left ideal, then either N = J(A), or N = 0 and A is a right uniform domain.

     

Keywords

Prime Ideal Left Ideal Division Ring Noetherian Ring Semiprime Ring 
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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Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Askar A. Tuganbaev
    • 1
  1. 1.Moscow Power Engineering InstituteTechnological UniversityMoscowRussia

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