Semiprimitive ring

In mathematics, especially in the area of algebra known as ring theory, a semiprimitive ring is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about the ring. Important 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. The quotient of every ring by its Jacobson radical is semiprimitive, allowing every ring to be understood to some extent through semiprimitive rings.

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, harv|Lam|1995|p=137.

A left artinian ring is semiprimitive if and only if it is semisimple, harv|Lam|2001|p=54.

Examples

* 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, harv|Jacobson|1989|p=203. 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, harv|Lam|1995|p=42.

References

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