central simple algebra meaning: definition, quotations

central simple algebra

noun

Algebra, Mathematics, Sciences A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K.

The complex numbers #92;C form a central simple algebra over themselves, but not over the real numbers #92;R (the centre of #92;C is all of #92;C, not just #92;R). The quaternions #92;mathbbH form a 4-dimensional central simple algebra over #92;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).

3 more examples

This crossed product E#92;alphaG was introduced by Noether and played a significant role in the classical theory of central simple algebras.1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products, Elsevier (North-Holland), 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.2007, Falko Lorenz, Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics, Springer, page 151:
Let A#95;1,A#95;2 be central simple algebras over a field K. Then A#95;1#92;otimes#95;KA#95;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#92;otimes#95;KA#92;operatorname#123;op#125;#92;cong#92;operatorname#123;Aut#125;#95;#123;K#92;operatorname#123;-Vect2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84: