Page "Central simple algebra" Paragraph 4

Given two central simple algebras A ~ M ( n, S ) and B ~ M ( m, T ) over the same field F, A and B are called similar ( or Brauer equivalent ) 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 ( F ) of the field F. It is always a torsion group.