Zorn's lemma is a form of the

axiom of choice which is technically very useful for proving

existence theorems. 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,

arguments of this

pattern are so common that they are referred to as

zornification or

zornication.

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).