Editing Semisimple module (section)

=== Jacobson semisimple ===
{{main|Semiprimitive ring}}
A ring is called ''Jacobson semisimple'' (or ''J-semisimple'' or ''[[semiprimitive ring|semiprimitive]]'') if the intersection of the maximal left ideals is zero, that is, if the [[Jacobson radical]] is zero.  Every ring that is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself.  A J-semisimple ring is semisimple if and only if it is an [[artinian ring]], so semisimple rings are often called ''artinian semisimple rings'' to avoid confusion.

For example, the ring of integers, '''Z''', is J-semisimple, but not artinian semisimple.