Quaternionic vector space

In mathematics, a left (or right) quaternionic vector space is a left (or right) $$\mathbb H$$-module where $$\mathbb H$$ is the division ring of quaternions. One must distinguish between left and right quaternionic vector spaces since $$\mathbb H$$ is non-commutative. Further, $$\mathbb H$$ is not a field, so quaternionic vector spaces are not vector spaces, but merely modules.

The space $$\mathbb H^n$$ is both a left and right quaternionic vector space using componentwise multiplication. Namely, for $$q \in \mathbb H$$ and $$(r_1, \ldots, r_n) \in \mathbb H^n$$,
 * $$ q (r_1, \ldots, r_n) = (q r_1, \ldots, q r_n),$$
 * $$ (r_1, \ldots, r_n) q = (r_1 q, \ldots, r_n q).$$

Since $$\mathbb H$$ is a division algebra, every finitely generated (left or right) $$\mathbb H$$-module has a basis, and hence is isomorphic to $$\mathbb H^n$$ for some $$n$$.