Radical of an ideal

In commutative ring theory, a branch of mathematics, the radical of an ideal formula_1 is an ideal such that an element formula_2 is in the radical if and only if some power of formula_2 is in formula_1 (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 non-commutative rings in the Semiprime ring article.
Definition.
The radical of an ideal formula_1 in a commutative ring formula_6, denoted by formula_7 or formula_8, is defined as
(note that formula_10).
Intuitively, formula_8 is obtained by taking all roots of elements of formula_1 within the ring formula_6. Equivalently, formula_8 is the pre-image of the ideal of nilpotent elements (the nilradical) in the quotient ring formula_15 (via the natural map formula_16). The latter shows formula_8 is itself an ideal.
If the radical of formula_1 is finitely generated, then some power of formula_8 is contained in formula_1. In particular, if formula_1 and formula_22 are ideals of a noetherian ring, then formula_1 and formula_22 have the same radical if and only if formula_1 contains some power of formula_22 and formula_22 contains some power of formula_1.
If an ideal formula_1 coincides with its own radical, then formula_1 is called a "radical ideal" or "semiprime ideal".
Properties.
This section will continue the convention that "I" is an ideal of a commutative ring formula_6:
Applications.
The primary motivation in studying radicals is "Hilbert's Nullstellensatz" in commutative algebra. One version of this celebrated theorem states that for any ideal formula_22 in the polynomial ring formula_119 over an algebraically closed field formula_120, one has
where
and
Geometrically, this says that if a variety formula_124 is cut out by the polynomial equations formula_125, then the only other polynomials which vanish on formula_124 are those in the radical of the ideal formula_127. 
Another way of putting it: the composition formula_128 is a closure operator on the set of ideals of a ring.