semiprimitive ring

A ring is said to be semiprimitive if its Jacobson radicalMathworldPlanetmath is the zero idealMathworldPlanetmathPlanetmath.

Any simple ringMathworldPlanetmath is automatically semiprimitive.

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

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

Note: The semiprimitive condition is sometimes also referred to as a semisimplePlanetmathPlanetmathPlanetmath, 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 ringsMathworldPlanetmath (

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