= 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.
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