Semiprime ideal - Knowlpedia

Let $R$ be a ring and let $I$ be a two-sided ideal. The ideal $I$ is semiprime if, for every two-sided ideal $J\subseteq R$ ,

J^2\subseteq I\quad\Longrightarrow\quad J\subseteq I.

Remarks

Equivalently, $R/I$ has no nonzero nilpotent two-sided ideals. In a commutative ring, semiprime ideals are exactly radical ideals.

Examples

In $k[x,y]$ , the ideal $(x)\cap(y)$ is semiprime because it is an intersection of prime ideals.
In $\mathbb Z$ , the ideal $(6)$ is semiprime because $6$ is squarefree.
The ideal $(4)\subseteq\mathbb Z$ is not semiprime, since $(2)^2\subseteq(4)$ but $(2)\nsubseteq(4)$ .