In mathematics, the Hurwitz quaternions are a subring of the ring H of all quaternions, consisting of all quaternions

a + bi + cj + dk

with a, b, c and d rational numbers, and either all integers, or all half-integers. It can be checked that this set of quaternions is closed under quaternion multiplication, so that it indeed forms a subring.

It is also a lattice in R⁴, and therefore forms an order in H, in the sense of ring theory. See F4 lattice. It is in fact a maximal order; and this accounts for its importance. It contains the order of quaternions with all the coefficients integers, which is the more obvious candidate for the idea of an integral quaternion. The latter subring has been called the Lipschitz quaternions. It is not a maximal order, and therefore (as it turns out) less suitable for developing a theory of left ideals comparable to that of algebraic number theory.

What Adolf Hurwitz realised, therefore, was that this definition of Hurwitz integral quaternion is the better one to operate with. This was one major step in the theory of maximal orders, the other being the remark that they will not in general, for a non-commutative ring such as H, be unique. One therefore needs to fix a maximal order to work with, in carrying over the concept of algebraic integer.