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 and only if some power of x is in I. (Taking the radical is called radicalization.) A radical ideal (or semiprime ideal) is an ideal that is equal to its own radical. The radical of a primary ideal is a prime 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 and only if some power of x is in I. (Taking the radical is called radicalization.) A radical ideal (or semiprime ideal) is an ideal that is equal to its own radical. The radical of a primary ideal is a prime ideal. This concept is 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 I {displaystyle {sqrt {I}}} , is defined as (Note that I ⊂ I {displaystyle Isubset {sqrt {I}}} .)Intuitively, I {displaystyle {sqrt {I}}} is obtained by taking all roots of elements of I within the ring R. Equivalently, I {displaystyle {sqrt {I}}} is the pre-image of the ideal of nilpotent elements (the nilradical) in the quotient ring R / I {displaystyle R/I} . The latter shows I {displaystyle {sqrt {I}}} is itself an ideal. If the radical of I is finitely generated, then some power of I {displaystyle {sqrt {I}}} is contained in I. In particular, if I and J are ideals of a noetherian ring, then I and J have the same radical if and only if I contains some power of J and J contains some power of I. If an ideal I coincides with its own radical, then I is called a radical ideal or semiprime ideal. This section will continue the convention that I is an ideal of a commutative ring R: where supp ⁡ M {displaystyle operatorname {supp} M} is the support of M and ass ⁡ M {displaystyle operatorname {ass} M} is the set of associated primes of M. The primary motivation in studying radicals is Hilbert's Nullstellensatz in commutative algebra. One version of this celebrated theorem states that for any ideal J in the polynomial ring k [ x 1 , x 2 , … , x n ] {displaystyle k} over an algebraically closed field k, one has