View source for Central simple algebra

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A central simple algebra is a simple associative algebra with a unit element that is a
[[Central algebra|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|Brauer group]] of $K$ relative to the operation induced
by tensor multiplication.

====References====
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|valign="top"|{{Ref|Al}}||valign="top"| A.A. Albert, "Structure of algebras", Amer. Math. Soc. (1939)  {{MR|0000595}}  {{ZBL|0023.19901}}    
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|valign="top"|{{Ref|De}}||valign="top"| M. Deuring, "Algebren", Springer (1935)   {{ZBL|0011.19801}}   {{MR|0228526}}  
|-
|valign="top"|{{Ref|DrKi}}||valign="top"| Yu.A. Drozd, V.V. Kirichenko, "Finite-dimensional algebras", Kiev (1980) (In Russian) {{MR|0591671}}  {{ZBL|0469.16001}}    
|-
|valign="top"|{{Ref|Pe}}||valign="top"| R.S. Peirce, "Associative algebras", Springer (1980) 
|-
|valign="top"|{{Ref|He}}||valign="top"| I.N. Herstein, "Noncommutative rings", Math. Assoc. Amer.  (1968)  {{MR|0227205}}  {{ZBL|0177.05801}}    
|-
|valign="top"|{{Ref|Ja}}||valign="top"| N. Jacobson, "Structure of rings", Amer. Math. Soc.  (1956)  {{MR|0081264}}  {{ZBL|0073.02002}}    
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