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.
Contents
Radical ideals defined here are generalized to noncommutative rings in the Semiprime ring article.
Definition
The radical of an ideal I in a commutative ring R, denoted by Rad(I) or
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
If the radical of I is finitely generated, then some power of
If an ideal I coincides with its own radical, then I is called a radical ideal or semiprime ideal.
Examples
Consider the ring Z of integers.
- The radical of the ideal 4Z of integer multiples of 4 is 2Z.
- The radical of 5Z is 5Z.
- The radical of 12Z is 6Z.
- In general, the radical of mZ is rZ, where r is the product of all distinct prime factors of m (each prime factor of m occurs exactly once as a factor of the product r) (see radical of an integer). In fact, this generalizes to an arbitrary ideal; see the properties section.
The radical of a primary ideal is prime. If the radical of an ideal I is maximal, then I is primary.
If I is an ideal,
Let I, J be ideals of a ring R. If
Let M be a finitely generated module over a noetherian ring R. Then
where
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 an algebraically closed field k, and for any finitely generated polynomial ideal J in the n indeterminates
where
and
Another way of putting it: The composition