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The Restricted Linear Group

  • F. E. A. JohnsonEmail author
Chapter
  • 942 Downloads
Part of the Algebra and Applications book series (AA, volume 17)

Abstract

A celebrated result of H.J.S. Smith (Jacobson in Basic algebra. Freeman, New York, 1974, Smith in Philos. Trans. 151:293:326, 1861) shows that when Λ is a commutative integral domain which possesses a Euclidean algorithm then an arbitrary m×m matrix X over Λ can be expressed as a product X=E + DE where D is diagonal and E +, E are products of elementary unimodular matrices. This chapter is a general study of rings whose matrices possess an analogue of such a Smith Normal Form.

Keywords

Smith Normal Form Commutative Integral Domain Strong Lifting Property Weak Determinant Surjective Ring Homomorphism 
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.

References

  1. 16.
    Cohn, P.M.: On the structure of the GL 2 of a ring. Publ. Math. IHES 30, 5–53 (1966) Google Scholar
  2. 17.
    Cohn, P.M.: Free Rings and Their Relations. LMS, 2nd edn. Academic Press, San Diego (1985) zbMATHGoogle Scholar
  3. 21.
    Dieudonné, J.: Les déterminants sur un corps noncommutatif. Bull. Soc. Math. Fr. 71, 27–45 (1943) zbMATHGoogle Scholar
  4. 47.
    Jacobson, N.: Basic Algebra. Freeman, New York (1974) zbMATHGoogle Scholar
  5. 64.
    Klingenberg, W.: Die Struktur der linearen Gruppe über einem nichtkommutativen lokalen Ring. Arch. Math. 13, 73–81 (1962) MathSciNetCrossRefGoogle Scholar
  6. 86.
    Smith, H.J.S.: On systems of linear indeterminate equations and congruences. Philos. Trans. 151, 293–326 (1861) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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