= 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.
Back to article page
