Prime Ideal - Important Facts

Important Facts

• Prime avoidance lemma: If R is a commutative ring, and A is a subring (possibly without unity), and I₁,...,I_n is a collection of ideals of R with at most two members not prime, then if A is not contained in any I_j, it is also not contained in the union of I₁,...,I_n. In particular, A could be an ideal of R.
• If S is any m-system in R, then a lemma essentially due to Krull shows that there exists an ideal of R maximal with respect to being disjoint from S, and moreover the ideal must be prime. In the case {S}={1}, we have Krull's theorem, and this recovers the maximal ideals of R. Another prototypical m-system is the set of all positive powers of a non-nilpotent element.
• For a prime ideal P, the complement R\P has another property beyond being an m-system. If xy is in R\P, then both x and y must be in R\P, since P is an ideal. A set which contains the divisors of its elements is called saturated.
• For a commutative ring R, there is a kind of converse for the previous statement: If S is any nonempty saturated and multiplicatively closed subset of R, the complement R\S is a union of prime ideals of R.
• The union and the intersection of a chain of prime ideals is a prime ideal. With Zorn's Lemma, this implies that the poset of prime ideals (partially ordered by inclusion) has maximal and minimal elements.