Artinian ring
Since a simple Artinian ring is a (full) matrix ring over a division ring, if "A" is a simple algebra, then "A" is a (full) matrix algebra over a division algebra "D" over "k"; i.e., formula_40. More generally, if "A" is a semisimple algebra, then it is a finite product of matrix algebras (over various division "k"-algebras), the fact known as the Artin–Wedderburn theorem.