Radical of an ideal

In ring theory, a branch of mathematics, the radical of an ideal of a commutative ring is another ideal defined by the property that an element is in the radical if and only if some power of is in. Taking the radical of an ideal is called radicalization. A radical 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 in a commutative ring, denoted by or, is defined as
.
Intuitively, is obtained by taking all roots of elements of within the ring. Equivalently, is the preimage of the ideal of nilpotent elements of the quotient ring . The latter proves that is an ideal.
If the radical of is finitely generated, then some power of is contained in. In particular, if and are ideals of a Noetherian ring, then and have the same radical if and only if contains some power of and contains some power of.
If an ideal coincides with its own radical, then is called a radical ideal or semiprime ideal.

Examples

Consider the ring of integers.
# The radical of the ideal of integer multiples of is .
# The radical of is.
# The radical of is.
# In general, the radical of is, where is the product of all distinct prime factors of, the largest square-free factor of . In fact, this generalizes to an arbitrary ideal.
Consider the ideal. It is trivial to show . Since is algebraically closed, every homomorphism must factor through, so we only have to compute the intersection of to compute the radical of We then find that

Properties

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

It is always true that, i.e. radicalization is an idempotent operation. Moreover, is the smallest radical ideal containing.
is the intersection of all the prime ideals of that contain and thus the radical of a prime ideal is equal to itself.
Specializing the last point, the nilradical is equal to the intersection of all prime ideals of This property is seen to be equivalent to the former via the natural map, which yields a bijection : defined by
An ideal in a ring is radical if and only if the quotient ring is reduced.
The radical of a homogeneous ideal is homogeneous.
The radical of an intersection of ideals is equal to the intersection of their radicals:.
The radical of a primary ideal is prime. If the radical of an ideal is maximal, then is primary.
If is an ideal,. Since prime ideals are radical ideals, for any prime ideal.
Let be ideals of a ring. If are comaximal, then are comaximal.
Let be a finitely generated module over a Noetherian ring. Then where is the support of and is the set of associated primes of.

Applications

One of the primary motivations for studying radicals of ideals is to understand algebraic sets and varieties in algebraic geometry.
For a subset of polynomials and subset of points, where is an algebraically [closed field], let
and
be the zero locus of S and vanishing ideal of X, respectively.
If is the ideal in generated by the elements of S, then. Moreover, the vanishing ideal is always a radical ideal:.
The operations V and I are, in a sense, inverses of each other:
For any subset of points X,, where is the closure of X in the Zariski topology. In particular, if X is an algebraic set, since algebraic sets are closed in the Zariski topology.
Hilbert's Nullstellensatz is a fundamental result in commutative algebra and algebraic geometry that addresses the composition of V and I in the opposite order. One version of this celebrated theorem states that for any ideal of polynomials, we have

As a corollary, if J is a radical ideal. Thus, we can state more precisely that the V and I operations give a bijective correspondence between radical ideals and algebraic sets: