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Concept
Radical of an ideal
Summary
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) 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.
The radical 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 of the ideal of nilpotent elements (the nilradical) 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.
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 (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 ).
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