# Semi-prime

A semi-prime is a term from abstract algebra . It represents an extension of the concept of the prime ideal .

## definition

In the following, let R be a ring with one . Then an ideal Q of R is a semi-prime ideal if it satisfies one of the following equivalent conditions:

• If R is an ideal with , then is .${\ displaystyle I \ triangleleft R}$ ${\ displaystyle I ^ {2} \ subseteq Q}$ ${\ displaystyle I \ subseteq Q}$ • Q is an average of prime ideals.

## properties

• A ring R is called semiprim, if it is semiprimid. Then the mapping , whereby the product is formed over all prime ideals, is injective. Therefore a semiprimer ring is a subdirect product of primer rings, that is, those in which the zero ideal is prime.${\ displaystyle \ {0 \}}$ ${\ displaystyle R \ rightarrow \ prod _ {P} R / P, \, x \ mapsto (x + P) _ {P}}$ • An average of semi-prime ideals is again a semi-prime ideal.
• The prime radical is the smallest semi- prime ideal.

## Individual evidence

1. Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), Theorem 2.6.17 applied to R / Q.
2. Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), § 2.2