Quaternionic vector space

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 H"n" of "n"-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication:
for quaternions "q" and "q"1, "q"2, ... "q""n".
Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to H"n" for some "n".