Important facts Prime avoidance lemma. If is a commutative ring, and is a subring (possibly without unity), and is a collection of ideals of with at most two members not prime, then if is not contained in any , it is also not contained in the union of .Jacobson Basic Algebra II, p. 390 In particular, could be an ideal of . If is any m-system in , then a lemma essentially due to Krull shows that there exists an ideal of maximal with respect to being disjoint from , and moreover the ideal must be prime (the primality can be proved as follows: if , then there exist elements such that by the maximal property of . Now, if , then , which is a contradiction). In the case we have Krull's theorem, and 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 .Kaplansky Commutative rings, p. 2 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.
Important facts Prime avoidance lemma. If is a commutative ring, and is a subring (possibly without unity), and is a collection of ideals of with at most two members not prime, then if is not contained in any , it is also not contained in the union of .Jacobson Basic Algebra II, p. 390 In particular, could be an ideal of . If is ,any m-system in , then a lemma essentially due to Krull shows that there exists an ideal of maximal with respect to being disjoint from , and moreover the ideal must be prime (the primality can be proved as follows: if , then there exist elements such that by the maximal property of . Now, if , then , which is a contradiction) In the case we have Krull's theorem, and 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 ideal.s of .Kaplansky Commutative rings, p. 2 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