Definition
The radical of an ideal I in a commutative ring R, denoted by Rad(I) or, is defined as
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 . The latter shows Rad(I) is an ideal itself, containing I.
If the radical of I is finitely generated, then for some n. In particular, If I and J are ideals of a noetherian ring, then I and J have the same radical if and only if I contains some power of J and J contains some power of I.
If an ideal I coincides with its own radical, then I is called a radical ideal or semiprime ideal.
Read more about this topic: Radical Of An Ideal
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