Editing Hurwitz quaternion (section)

{{Short description|Generalization of algebraic integers}}
In [[mathematics]], a '''[[Adolf Hurwitz|Hurwitz]] quaternion''' (or '''Hurwitz integer''') is a [[quaternion]] whose components are ''either'' all [[integer]]s ''or'' all [[half-integer]]s (halves of [[parity (mathematics)|odd]] integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz quaternions is

:<math>H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \;\mbox{ or }\, a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.</math>

That is, either ''a'', ''b'', ''c'', ''d'' are all integers, or they are all half-integers.
''H'' is closed under quaternion multiplication and addition, which makes it a [[subring]] of the [[ring (mathematics)|ring]] of all quaternions '''H'''. Hurwitz quaternions were introduced by {{harvs|txt|last=Hurwitz|first=Adolf|authorlink=Adolf Hurwitz|year=1919}}.

A '''Lipschitz quaternion''' (or '''Lipschitz integer''') is a quaternion whose components are all integers. The set of all Lipschitz quaternions

:<math>L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}</math>

forms a subring of the Hurwitz quaternions ''H''. Hurwitz integers have the advantage over Lipschitz integers that it is possible to perform [[Euclidean division]] on them, obtaining a small remainder.

Both the Hurwitz and Lipschitz quaternions are examples of [[noncommutative ring|noncommutative]] [[domain (ring theory)|domains]] which are not [[division ring|division rings]].