is a form of the axiom of choice
which is technically very useful for proving existence theorem
s. For instance, it follows directly from Zorn's lemma
that every ring
has a maximal ideal
and every vector space
has a basis
(algebraic, that is, Hamel basis
). In some subfields of mathematics
s of this pattern
are so common that they are referred to as zornification
To rephrase BelDion's statement above a little: In a poset, if every chain has an upper bound, then the entire poset has a maximal element.
Some Polish mathematicians refer to this lemma as the Kuratowski-Zorn lemma, to properly credit its first appearance, in a paper of Kazimierz Kuratowski. For more information see Set theory for the working mathematician by Krzyzstof Ciesielski (London Mathematical Society student texts, Cambridge University Press).