Radical of an ideal

In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal (or semiprime ideal) is an ideal that is its own radical (this can be phrased as being a fixed point of an operation on ideals called 'radicalization'). The radical of a primary ideal is prime.

Radical ideals defined here are generalized to noncommutative rings in the Semiprime ring article.

The radical of an ideal I in a commutative ring R, denoted by Rad(I) or ${\displaystyle {\sqrt {I}}}$, is defined as

Intuitively, one can think of the radical of I as obtained by taking all the possible roots of elements of I. Equivalently, the radical of I is the pre-image of the ideal of nilpotent elements (called nilradical) in ${\displaystyle R/I}$. The latter shows ${\displaystyle {\sqrt {I}}}$ is an ideal itself, containing I.

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