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, (Lam 1995, p. 137).
A left artinian ring is semiprimitive if and only if it is semisimple, (Lam 2001, p. 54). Such rings are sometimes called semisimple Artinian, (Kelarev 2002, p. 13).
Read more about this topic: Semiprimitive Ring
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