Semiprime ring

A Semiprimideal is a term used in abstract algebra. It represents an extension of the notion of prime ideal

Definition

In the following, let R be a ring with unity. Then Q is an ideal of R is a Semiprimideal if it meets one of the following equivalent conditions:

• Is an ideal of R with, then.
• Q is an average of prime ideals.

Properties

• A ring R is called semiprime if a Semiprimideal is. Then the map, where the product is taken over all prime ideals injective. Therefore, a semi- ring is a subdirect product primer primer rings, ie those in which the zero ideal is prime.
• An average of Semiprimidealen is a Semiprimideal again.
• The Primradikal is the smallest Semiprimideal.