semiprime idealSemiprimeIdeal2001-11-23 21:59:002003-11-29 21:34:01Definitionsemiprime ringsemiprime\usepackage{amssymb}
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\usepackage{xypic}Let $R$ be a ring.
An ideal $I$ of $R$ is a {\it 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 {\it 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.