In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in 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.

Definition

The radical of an ideal I in a commutative ring R, denoted by or \sqrt{I}, is defined as (note that ). Intuitively, \sqrt{I} is obtained by taking all roots of elements of I within the ring R. Equivalently, \sqrt{I} is the preimage of the ideal of nilpotent elements (the nilradical) of the quotient ring R/I (via the natural map ). The latter proves that \sqrt{I} is an ideal. If the radical of I is finitely generated, then some power of \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.

Examples

Properties

This section will continue the convention that I is an ideal of a commutative ring R:

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 J in the polynomial ring over an algebraically closed field \mathbb{k}, one has where and Geometrically, this says that if a variety V is cut out by the polynomial equations, then the only other polynomials that vanish on V are those in the radical of the ideal. Another way of putting it: the composition is a closure operator on the set of ideals of a ring.

Citations