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In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about the ring. Rings such as the ring of integers are semiprimitive, and an artinian semiprimitive ring is just a semisimple ring. Semiprimitive rings can be understood as subdirect products of primitive rings, which are described by the Jacobson density theorem. ==Definition== A ring is called semiprimitive or Jacobson semisimple if its Jacobson radical is the zero ideal. A ring is semiprimitive if and only if it has a faithful semisimple left module. The semiprimitive property is left-right symmetric, and so a ring is semiprimitive if and only if it has a faithful semisimple right module. A ring is semiprimitive if and only if it is a subdirect product of left primitive rings. A commutative ring is semiprimitive if and only if it is a subdirect product of fields, . A left artinian ring is semiprimitive if and only if it is semisimple, . Such rings are sometimes called semisimple Artinian, . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Semiprimitive ring」の詳細全文を読む スポンサード リンク
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