Semisimple ring

A semisimple ring is a unital ring $R$ such that every left $R$-module is semisimple (i.e. a direct sum of simple modules). Equivalently, the left regular module ${}_RR$ is a direct sum of minimal left ideals.

In structural terms, semisimple rings are exactly those described by the Artin–Wedderburn theorem : they are finite products of matrix rings over division rings . A key obstruction to semisimplicity is the Jacobson radical , which vanishes for semisimple rings.

Examples:

• $M_n(k)$ is semisimple for any field $k$ and integer $n\ge 1$.
• $M_2(\mathbb{Q})\times \mathbb{Q}$ is semisimple.
• The ring $k[x]/(x^2)$ is not semisimple: the class of $x$ is nonzero but nilpotent, forcing a nontrivial Jacobson radical.