central simple algebra

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central simple algebra

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central simple algebra (plural central simple algebras)

(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 $\mathbb {C}$ form a central simple algebra over themselves, but not over the real numbers $\mathbb {R}$ (the centre of $\mathbb {C}$ is all of $\mathbb {C}$ , not just $\mathbb {R}$ ). The quaternions $\mathbb {H}$ form a 4-dimensional central simple algebra over $\mathbb {R}$ .

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 $E^{\alpha }G$ 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 $D$ .
- 2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84:
  Let $A_{1},A_{2}$ be central simple algebras over a field $K$ . Then $A_{1}\otimes _{K}A_{2}$ can be shown to be a central simple algebra over $K$ . Further, if $A$ is a central simple algebra over a field $K$ , then $A\otimes _{K}A^{\operatorname {op} }\cong \operatorname {Aut} _{K\operatorname {-Vect} }(A)$ . I.e., it is isomorphic to a matrix algebra.