# Central simple algebra

2010 Mathematics Subject Classification: *Primary:* 16-XX [MSN][ZBL]

A central simple algebra is a simple associative algebra with a unit element that is a
central algebra. Every finite-dimensional central
simple algebra $A$ over a field $K$ is isomorphic to a matrix algebra
$M_n(C)$ over a finite-dimensional central division algebra $C$ over
$K$. In particular, if $K$ is algebraically closed, then every
finite-dimensional central simple algebra $A$ over $K$ is isomorphic
to $M_n(K)$, and if $K=\R$, then $A$ is isomorphic to the algebra of real or
quaternion matrices. The tensor product of a central simple algebra
$A$ and an arbitrary simple algebra $B$ is a simple algebra, which is
central if $B$ is central. Two finite-dimensional central simple
algebras $A$ and $B$ over $K$ are called equivalent if
$$A\otimes_K M_m(K) \cong B\otimes M_n(K)$$
for
certain $m$ and $n$, or, which is equivalent, if $A$ and $B$ are
isomorphic matrix algebras over one and the same central division
algebra. The equivalence classes of central simple algebras over $K$
form the
Brauer group of $K$ relative to the operation induced
by tensor multiplication.

#### References

[Al] | A.A. Albert, "Structure of algebras", Amer. Math. Soc. (1939) MR0000595 Zbl 0023.19901 |

[De] | M. Deuring, "Algebren", Springer (1935) Zbl 0011.19801 MR0228526 |

[DrKi] | Yu.A. Drozd, V.V. Kirichenko, "Finite-dimensional algebras", Kiev (1980) (In Russian) MR0591671 Zbl 0469.16001 |

[Pe] | R.S. Peirce, "Associative algebras", Springer (1980) |

[He] | I.N. Herstein, "Noncommutative rings", Math. Assoc. Amer. (1968) MR0227205 Zbl 0177.05801 |

[Ja] | N. Jacobson, "Structure of rings", Amer. Math. Soc. (1956) MR0081264 Zbl 0073.02002 |

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Central simple algebra.

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