Radical of an ideal

In ring theory, a branch of mathematics, the radical of an ideal ${\displaystyle I}$ of a commutative ring is another ideal defined by the property that an element ${\displaystyle x}$ is in the radical if and only if some power of ${\displaystyle x}$ is in ${\displaystyle I}$. Taking the radical of an ideal is called radicalization. A radical ideal (or semiprime ideal) is an ideal that is equal to its radical. The radical of a primary ideal is a prime ideal.

This concept is generalized to non-commutative rings in the semiprime ring article.