Editing Prime ideal (section)

==Important facts==
*'''[[Prime avoidance lemma]].''' If {{mvar|R}} is a commutative ring, and {{mvar|A}} is a [[subring]] (possibly without unity), and {{math|''I''<sub>1</sub>, ..., ''I<sub>n</sub>''}} is a collection of ideals of {{mvar|R}} with at most two members not prime, then if {{mvar|A}} is not contained in any {{math|''I<sub>j</sub>''}}, it is also not contained in the [[union (set theory)|union]] of {{math|''I''<sub>1</sub>, ..., ''I<sub>n</sub>''}}.<ref>Jacobson ''Basic Algebra II'', p. 390</ref> In particular, {{mvar|A}} could be an ideal of {{mvar|R}}.
* If {{mvar|S}} is any m-system in {{mvar|R}}, then a lemma essentially due to Krull shows that there exists an ideal {{mvar|I}} of {{mvar|R}} maximal with respect to being disjoint from {{mvar|S}}, and moreover the ideal {{mvar|I}} must be prime (the primality {{mvar|I}} can be [[mathematical proof|proved]] as follows: if <math>a, b\not\in I</math>, then there exist elements <math>s, t\in S</math> such that <math>s\in I+(a), t\in I+(b)</math> by the maximal property of {{mvar|I}}. Now, if <math>(a)(b)\subset I</math>, then <math>st\in (I+(a))(I+(b))\subset I+(a)(b)\subset I</math>, which is a contradiction).<ref name="Lam"/> In the case {{math|''S'' {{=}} {1},}} we have [[Krull's theorem]], and this recovers the maximal ideals of {{mvar|R}}. Another prototypical m-system is the set, {{math|{''x'', ''x''<sup>2</sup>, ''x''<sup>3</sup>, ''x''<sup>4</sup>, ...},}} of all positive powers of a non-[[nilpotent]] element.
* For a prime ideal {{mvar|P}}, the complement {{math|''R''∖''P''}} has another property beyond being an m-system. If ''xy'' is in {{math|''R''∖''P''}}, then both {{mvar|x}} and {{mvar|y}} must be in {{math|''R''∖''P''}}, since {{mvar|P}} is an ideal. A set that contains the divisors of its elements is called '''saturated'''.
* For a commutative ring {{mvar|R}}, there is a kind of converse for the previous statement: If {{mvar|S}} is any nonempty saturated and multiplicatively closed subset of {{mvar|R}}, the complement {{math|''R''∖''S''}} is a union of prime ideals of {{mvar|R}}.<ref>Kaplansky ''Commutative rings'', p. 2</ref>
*The [[intersection (set theory)|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.