Simple algebra

A simple unital associative algebra is nonzero and has no nonzero proper two-sided ideals. Finite-dimensional complex examples are exactly the full matrix algebras $M_d(\mathbb{C})$, by the Artin–Wedderburn theorem. A simple associative algebra need not be a division algebra: for $d>1$ it has noninvertible nonzero matrices. Any nonzero two-sided ideal of $M_d(\mathbb{C})$ contains a matrix unit, by multiplying a nonzero entry on the left and right by suitable matrix units, and therefore contains all of them.