In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H denotes the noncommutative ring of the quaternions. The space Hn of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication: for quaternions q and q1, q2, ... qn. Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to Hn for some n.