Semiprimitive ring
Template:Ring theory sidebar 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.
DefinitionEdit
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, Template:Harv.
A left artinian ring is semiprimitive if and only if it is semisimple, Template:Harv. Such rings are sometimes called semisimple Artinian, Template:Harv.
ExamplesEdit
- The ring of integers is semiprimitive, but not semisimple.
- Every primitive ring is semiprimitive.
- The product of two fields is semiprimitive but not primitive.
- Every von Neumann regular ring is semiprimitive.
Jacobson himself has defined a ring to be "semisimple" if and only if it is a subdirect product of simple rings, Template:Harv. However, this is a stricter notion, since the endomorphism ring of a countably infinite dimensional vector space is semiprimitive, but not a subdirect product of simple rings, Template:Harv.