A ring is said to be semiprimitive if its Jacobson radical is the zero ideal.

Any simple ring is automatically semiprimitive.

A finite direct product of matrix rings over division rings can be shown to be semiprimitive and both left and right Artinian.

The Artin-Wedderburn Theorem states that any semiprimitive ring which is left or right Artinian is isomorphic to a finite direct product of matrix rings over division rings.

Note: The semiprimitive condition is sometimes also referred to as a semisimple, Jacobson semisimple, or J-semisimple. Furthermore, when either of the last two names are used, the adjective 'semisimple' is frequently intended to refer to a ring that is semiprimitive and Artinian (see the entry on semisimple rings).