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In ring theory and related areas of mathematics a central simple algebra over a field K is a finite-dimensional associative K-algebra A which is simple, and for which the center is exactly K. {\displaystyle K} is a simple algebra with center K, but is not a central simple algebra over K as it has infinite dimension as a K-module.]
For example, the complex numbers C form a CSA over themselves, but not over the real numbers R. The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals.
Given two central simple algebras A ~ M and B ~ M over the same field F, A and B are called similar if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br of the field F. It is always a torsion group.