Semiprime ideal

Let $R$ be a ring and let $I$ be a two-sided ideal . The ideal $I$ is semiprime if for every two-sided ideal $J\subseteq R$, the condition $J^2\subseteq I$ implies $J\subseteq I$.

Equivalently, the quotient $R/I$ has no nonzero nilpotent ideals, i.e. it contains no nonzero ideal all of whose elements are nilpotent . In a commutative ring, semiprime ideals are exactly radical ideals .

Examples:

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