# Central simple algebra

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

A central simple algebra is a simple associative algebra with a unit element (cf. Simple algebra, 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
How to Cite This Entry:
Central simple algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Central_simple_algebra&oldid=21552
This article was adapted from an original article by r group','../u/u095350.htm','Unitary transformation','../u/u095590.htm','Vector bundle, analytic','../v/v096400.htm')" style="background-color:yellow;">A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article