central simple algebra

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English

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Noun

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central simple algebra (plural central simple algebras)

  1. (algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K.
    The complex numbers form a central simple algebra over themselves, but not over the real numbers (the centre of is all of , not just ). The quaternions form a 4-dimensional central simple algebra over .
    The concept of central simple algebra over a field K represents a noncommutative analogue to that of extension field over K. In both cases, the object has no nontrivial two-sided ideals and has a distinguished field in its centre, although a central simple algebra need not be commutative and need not have inverses (does not have be a division algebra).
    • 1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products[1], Elsevier (North-Holland), page 151:
      This crossed product was introduced by Noether and played a significant role in the classical theory of central simple algebras.
    • 2007, Falko Lorenz, Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics, Springer, page 151:
      Because of Wedderburn's theorem it is natural to call two central-simple algebras similar if they are isomorphic to matrix algebras over the same division algebra .
    • 2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84:
      Let be central simple algebras over a field . Then can be shown to be a central simple algebra over . Further, if is a central simple algebra over a field , then . I.e., it is isomorphic to a matrix algebra.

Synonyms

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  • CSA (initialism)

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