Krull's Theorem
From AoPSWiki
Krull's theorem is a result in ring theory. It states every ring has a maximal ideal. It is equivalent to the axiom of choice. It is named for Wolfgang Krull, who stated the theorem first, in 1929.
Full statement and proof
Theorem. Let 
be a ring, and let 
be a (left, right, two-sided) ideal of that is distinct from . Then there exists a maximal (left, right, two-sided) ideal of containing .
Proof. Note that an ideal of is distinct from if and only if it does not contain 1. Let 
be the family of proper ideals of containig . Evidently, 1 is not an element of any member of this family, so the union of a totally ordered subset of this family is an element of the family. It then follows from Zorn's Lemma that has a maximal element. 
