semiprime ideal

Let $R$ be a ring. An ideal $I$ of $R$ is a semiprime ideal if it satisfies the following equivalent conditions:

(a) $I$ can be expressed as an intersection of prime ideals of $R$ ;

(b) if $x \in R$ , and $xRx \subset I$ , then $x \in I$ ;

(c) if $J$ is a two-sided ideal of $R$ and $J^2 \subset I$ , then $J \subset I$ as well;

(d) if $J$ is a left ideal of $R$ and $J^2 \subset I$ , then $J \subset I$ as well;

(e) if $J$ is a right ideal of $R$ and $J^2 \subset I$ , then $J \subset I$ as well.

Here $J^2$ is the product of ideals $J \cdot J$ .

The ring $R$ itself satisfies all of these conditions (including being expressed as an intersection of an empty family of prime ideals) and is thus semiprime.

A ring $R$ is said to be a semiprime ring if its zero ideal is a semiprime ideal.

Note that an ideal $I$ of $R$ is semiprime if and only if the quotient ring $R/I$ is a semiprime ring.