A theorem on simple algebras
J. H. M. Wedderburn · Bulletin of the American Mathematical Society · 1925
In a previous paper 11 showed that every simple algebra A can be expressed as the direct product of a division algebra D and a simple matric algebra M=(e pq ); the object of this note is to show that this expression is unique, that is, if A = DiXMx = D 2 xM 2 , ivhere D x and D 2 are division algebras and M± and M 2 are simple matric algebras, then Dx and M ± are simply isomorphic% with D 2 and M 2 respectively.Let d ± and S 2 be the orders of D ± and D 2 , and let e x and e 2 be primitive idempotent elements of M t and M 2 respectively.If e x and e 2 are supplementary or equal, then § A ^ 6 t Ae 1 ^ e 2 Ae 2 ^ D 2 ;