this recovers the maximal ideals of . Another prototypical m-system is the set, of all positive powers of a non-nilpotent element. For a prime ideal , the complement has another property beyond being an m-system. If xy is in , then both and must be in , since is an ideal. A set that contains the divisors of its elements is called saturated. For a commutative ring , there is a kind of converse for the previous statement: If is any nonempty saturated and multiplicatively closed subset of , the complement is a union of prime ideals of . The intersection of members of a descending chain of prime ideals is a prime ideal, and in a commutative ring the union of members of an ascending chain of prime ideals is a prime ideal. With Zorn's Lemma, these observations imply that the poset of prime ideals of a commutative ring (partially ordered by inclusion) has maximal and minimal elements. Connection to maximality Prime ideals can frequently be produced as maximal elements of certain collections of ideals. For example: An ideal maximal with respect to having empty intersection with a fixed m-system is prime. An ideal maximal among annihilators of submodules of a fixed -module is prime. In a commutative ring, an ideal maximal with respect to being non-principal is prime. In a commutative ring, an ideal maximal with respect to being not countably generated is prime. See also Radical ideal Maximal ideal References Further reading