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 , by the Artin–Wedderburn theorem. A simple associative algebra need not be a division algebra: for it has noninvertible nonzero matrices. Any nonzero two-sided ideal of contains a matrix unit, by multiplying a nonzero entry on the left and right by suitable matrix units, and therefore contains all of them.
New to topics? Read the docs here!