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Applied Categorical Structures

, Volume 3, Issue 2, pp 139–149 | Cite as

Ideals, radicals, and structure of additive categories

  • Ross Street
Article

Abstract

Simple and semisimple additive categories are studied. We prove, for example, that an artinian additive category is (semi)simple iff it is Morita equivalent to a division ring(oid). Semiprimitive additive categories (that is, those with zero radical) are those which admit anoether full, faithful functor into a category of modules over a division ringoid.

Mathematics subject classifications (1991)

18E05 16A20 16A40 

Key words

Additive category semiprimitive ring artinian simple Jacobson radical density arguments 

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References

  1. 1.
    Karlheinz Baumgartner: Structure of additive categories,Cahiers topologie et géométrie différentielle 16 (1975), 169–213.Google Scholar
  2. 2.
    Karlheinz Baumgartner: General Wedderburn theorems and density for small categories (Preprint, Voitsberg, Austria, March 1987).Google Scholar
  3. 3.
    H. Cartan and S. Eilenberg:Homological Algebra, Princeton, 1956.Google Scholar
  4. 4.
    B. J. Day and R. H. Street: Categories in which all strong generators are dense,J. Pure Appl. Algebra 43 (1986), 235–242.Google Scholar
  5. 5.
    I. N. Hernstein:Noncommutative Rings, The Carus Math Monographs 15, Math. Assoc. of America, 1968.Google Scholar
  6. 6.
    Nathan Jacobson:Basic Algebra II, W. H. Freeman & Co., San Francisco, 1980.Google Scholar
  7. 7.
    G. M. Kelly: On the radical of a category,J. Australian Math. Soc. 4 (1964), 299–307.Google Scholar
  8. 8.
    Barry Mitchell: Rings with several objects,Advances in Math. 8 (1972), 1–161.Google Scholar
  9. 9.
    Ross Street: Enriched categories and cohomology,Quaestiones Math. 6 (1983), 265–283.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Ross Street
    • 1
  1. 1.Mathematics DepartmentMacquarie UniversityAustralia

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