# 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 |