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Difference between revisions of "Central simple algebra"

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A central simple algebra is a simple associative algebra with a unit element that is a
 
A central simple algebra is a simple associative algebra with a unit element that is a
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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|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}}  
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|valign="top"|{{Ref|De}}||valign="top"| M. Deuring, "Algebren", Springer (1935)  {{ZBL|0011.19801}}   {{MR|0228526}} 
 
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|valign="top"|{{Ref|DrKi}}||valign="top"| Yu.A. Drozd, V.V. Kirichenko, "Finite-dimensional algebras", Kiev (1980) (In Russian) {{MR|0591671}}  {{ZBL|0469.16001}}  
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|valign="top"|{{Ref|DrKi}}||valign="top"| Yu.A. Drozd, V.V. Kirichenko, "Finite-dimensional algebras", Kiev (1980) (In Russian) {{MR|0591671}}  {{ZBL|0469.16001}}  
 
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|valign="top"|{{Ref|Pe}}||valign="top"| R.S. Peirce, "Associative algebras", Springer (1980)  
 
|valign="top"|{{Ref|Pe}}||valign="top"| R.S. Peirce, "Associative algebras", Springer (1980)  
 
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|valign="top"|{{Ref|He}}||valign="top"| I.N. Herstein, "Noncommutative rings", Math. Assoc. Amer.  (1968)  {{MR|0227205}}  {{ZBL|0177.05801}}  
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|valign="top"|{{Ref|He}}||valign="top"| I.N. Herstein, "Noncommutative rings", Math. Assoc. Amer.  (1968)  {{MR|0227205}}  {{ZBL|0177.05801}}  
 
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|valign="top"|{{Ref|Ja}}||valign="top"| N. Jacobson, "Structure of rings", Amer. Math. Soc.  (1956)  {{MR|0081264}}  {{ZBL|0073.02002}}  
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|valign="top"|{{Ref|Ja}}||valign="top"| N. Jacobson, "Structure of rings", Amer. Math. Soc.  (1956)  {{MR|0081264}}  {{ZBL|0073.02002}}  
 
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Revision as of 21:42, 5 March 2012

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
How to Cite This Entry:
Central simple algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Central_simple_algebra&oldid=21373
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