Page "Semisimple module" Paragraph 23

A ring is called Jacobson semisimple ( or J-semisimple or semiprimitive ) if the intersection of the maximal left ideals is zero, that is, if the Jacobson radical is zero.

Every ring which 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.