# semiprimitive ring

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 (http://planetmath.org/WedderburnArtinTheorem) 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 (http://planetmath.org/SemisimpleRing2)).

 Title semiprimitive ring Canonical name SemiprimitiveRing Date of creation 2013-03-22 12:36:14 Last modified on 2013-03-22 12:36:14 Owner yark (2760) Last modified by yark (2760) Numerical id 20 Author yark (2760) Entry type Definition Classification msc 16N20 Synonym semisimple ring Synonym Jacobson semisimple ring Synonym J-semisimple ring Synonym semi-primitive ring Synonym semi-simple ring Synonym Jacobson semi-simple ring Synonym J-semi-simple ring Related topic SemisimpleRing2 Related topic WedderburnArtinTheorem Defines semiprimitivity Defines semiprimitive Defines semisimple Defines Jacobson semisimple Defines J-semisimple Defines semi-primitivity Defines semi-primitive Defines semi-simple Defines Jacobson semi-simple Defines J-semi-simple