Algebras and Representation Theory

, Volume 10, Issue 3, pp 257–270 | Cite as

The Prime Spectrum and the Extended Prime Spectrum of Noncommutative Rings

  • Igor KlepEmail author
  • Marcus Tressl


We investigate the prime spectrum of a noncommutative ring and its spectral closure, the extended prime spectrum. We construct a ring for which the prime spectrum is a spectral space different from the extended prime spectrum and we construct a von Neumann regular ring for which the prime spectrum is not a spectral space.

Key words

prime ideals spectral spaces 

Mathematics Subject Classifications (2000)

06E15 16D25 


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  1. 1.
    Belluce, L.P.: Spectral spaces and noncommutative rings. Comm. Algebra 19, 1855–1865 (1991)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Belluce, L.P.: Spectral closure for non-commutative rings. Comm. Algebra 25, 1513–1536 (1997)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cohn, P.M.: On n-simple rings. Algebra Universalis 53, 301–305 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Goodearl, K.R.: Von Neumann Regular Rings. Pitman, London (1979)zbMATHGoogle Scholar
  5. 5.
    Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser Verlag, Basel, (1998)zbMATHGoogle Scholar
  6. 6.
    Hochster, M.: Prime ideal structure in commutative rings. Trans. Am. Math. Soc. 142, 43–60 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kaplansky, I.: Topics in commutative ring theory, Dept. of Math., University of ChicagoGoogle Scholar
  8. 8.
    Lam, T.Y.: A first course in noncommutative rings, 2nd edn. Grad. Texts in Math. 131, Springer, Berlin Heidelberg New York (2001)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Ljubljana, IMFMLjubljanaSlovenia
  2. 2.Universität Regensburg, NWF I-MathematikRegensburgGermany

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