Primitive ring

Summary

A left primitive ring is a ring that has a faithful simple left module. By the Jacobson density theorem, such a ring acts densely on a vector space over a division ring. So it is close to a full ring of linear maps. Every left primitive ring is prime, and a commutative ring is primitive exactly when it is a field. Primitive rings are the building blocks of Jacobson's structure theory: every semiprimitive ring is a subdirect product of primitive rings.