Difference between revisions of "Krull's Theorem"
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Latest revision as of 08:38, 16 June 2008
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.