In ring theory, a branch of mathematics, the radical of an ideal of a commutative ring is in the radical if and only if. 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 in a commutative ring, is defined as

within the ring is the preimage of the ideal of nilpotent elements (the nilradical of the ring) of the quotient ring is an ideal.

is finitely generated. are ideals of a Noetherian ring.

is called a radical ideal or semiprime ideal.

Examples

of integers.

1. (the evens).

2. .

3. .

4. is the product of all distinct prime factors, the largest square-free (see Radical of an integer). In fact, this generalizes to an arbitrary ideal (see the Properties section).

5. (using the basic property but we give some alternative methods: corresponds to the nilradical, 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 is algebraically closed

Properties

:

• , i.e. radicalization is an idempotent.

• is the intersection of all the prime idealsand thus the radical of a prime ideal is equal to itself.

• Specializing the last point, the nilradical, which yields a bijection

• is radical if and only if the quotient ring is reduced.

• The radical of a homogeneous ideal is homogeneous.

• .

• The radical of a primary ideal is primary.

• .

• are comaximal are comaximal.

• be a finitely generated module over a Noetherian ring. Then is the support is the set of associated primes.

Applications to algebraic geometry

One of the primary motivations for studying radicals of ideals is to understand algebraic sets and varieties in algebraic geometry.

is an algebraically closed field, let

and

be the zero locus (or 'variety') of S and vanishing ideal (or ideal') of X, respectively.

is the ideal generated by the elements of S is the radical of that ideal, then these collections of polynomials have the same zero loci:

is always a radical ideal:

The operations V and I are, in a sense made more precise below, inverses of each other:

,

is the closure of X in the Zariski topology 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 we have

Geometrically, this says that if an affine algebraic set W is cut out by the polynomial equations then the polynomials f that vanish on W are precisely those such that one of the powers of f for some natural number r if J is a radical ideal.

and so V and I form an antitone Galois connection that gives rise to a bijective correspondence when restricted to radical ideals and algebraic sets:

See also

• Jacobson radical

• Nilradical of a ring

• Real radical

Notes

Citations

References