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 (or semiprime ideal or reduced 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 (occasionally also called the nilradical) of an ideal in a commutative ring , denoted by or , is defined as

(note that ). Intuitively, is obtained by taking all roots of elements of within the ring . Equivalently, is the preimage of the ideal of nilpotent elements (the nilradical of the ring) of the quotient ring (via the natural map ). 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 evens).
• # 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 (see Radical of an integer). In fact, this generalizes to an arbitrary ideal (see the Properties section).
• Consider the ideal . It is trivial to show (using the basic property but we give some alternative methods: The radical corresponds to the nilradical of the quotient ring , which is the intersection of all prime ideals of the quotient ring. This is contained in the Jacobson radical, which is the intersection of all maximal ideals, which are the kernels of homomorphisms to fields. Any ring homomorphism must have in the kernel in order to have a well-defined homomorphism (if we said, for example, that the kernel should be the composition of would be , which is the same as trying to force ). 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 (the set of all nilpotent elements) 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 (or '<nowiki/>variety<nowiki/>') of S and vanishing ideal (or ideal<nowiki/>') of X, respectively.