intersection of to compute the radical of We then find that Properties This section will continue the convention that I 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. Proof: On one hand, every prime ideal is radical, and so this intersection contains . Suppose is an element of which is not in , and let be the set . By the definition of , must be disjoint from . is also multiplicatively closed. Thus, by a variant of Krull's theorem, there exists a prime ideal that contains and is still disjoint from (see Prime ideal). Since contains , but not , this shows that is not in the intersection of prime ideals containing . This finishes the proof. The statement may be strengthened a bit: the radical of is the intersection of all prime ideals of that are minimal among those containing . 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 The primary motivation in studying radicals is Hilbert's Nullstellensatz in commutative algebra. One version of this celebrated theorem states that for any ideal in the polynomial ring over an algebraically closed field , one has where and Geometrically, this says that if a variety is cut out by the polynomial equations , then the only other