Radical of An Ideal - Properties

Properties

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

• It is always true that Rad(Rad(I))=Rad(I). In words, this says that Rad(I) is the smallest radical ideal containing I.

• Rad(I) is the intersection of all the prime ideals of R that contain I. On one hand, every prime ideal is radical, and so the intersection J of the prime ideals containing I contains Rad(I). Suppose r is an element of R which is not in Rad(I), and let S be the set {rn|n is a nonnegative integer}. By the definition of Rad(I), S must be disjoint from I. Since S is multiplicatively closed and R has identity, Zorn's lemma says that there exists an ideal P in this ring which contains I and is maximal with respect to being disjoint from S. It can be shown that P is a prime ideal. Since P contains I, but not r, this shows that r is not in the intersection of prime ideals containing I. Thus, the intersection of prime ideals containing I is contained in Rad(I), proving equality.

• Specializing the last point, the nilradical (the set of all nilpotent elements) is equal to the intersection of all prime ideals of R.

• An ideal I in a ring R is radical if and only if the quotient ring R/I is reduced.

• The radical of a homogeneous ideal is homogeneous.