Remark 1

Let K be a commutative field, and let A be a semisimple algebra over K. Then there exist K-algebras $D_1, . . . ,D_s$ that are fields and integers $r_1\geqslant 1, . . . , r_s\geqslant 1$ such that the K-algebra A is isomorphic to the product $\prod^s_{i=1}\mathbf{M}_{r_i}(D_i)$.