Semisimplicity of a right Artinian ring with zero radical

The finite intersections of maximal right ideals of a right Artinian ring have a minimal member by the descending chain condition. Intersecting it with any other maximal right ideal changes nothing, so it equals the Jacobson radical. If that radical is zero, the diagonal map from the regular module into a finite direct sum of its simple quotients is injective. A submodule of a finite direct sum of simple modules is semisimple and has finite length. This proves semisimplicity of the regular module without assuming Noetherianity first.