Central simple algebra

2020 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