# Vector space versus Free module

A vector space is simple a module over a field as opposed to an arbitrary ring. But, there are times, for example when $$R=D$$ a division ring where we refer to a module as a vector space. For instance, $$M_n(D)$$ as a vector space of dimension $$n^2$$ over $$D$$. However, this clearly would not satisfy the definition of a vector space for an arbitrary division ring since the scalar addition may not necessarily be commutative.

In instances such as these, are authors just abusing notation and really just mean a free module? To generalize the example above, $$M_n(R)$$ for an arbitrary ring is of dimension $$n^2$$ over $$R$$ and, hence, is a free object over (which acts on another free module $$R^n$$) $$R$$.

## 1 Answer

The term "vector space" is frequently used for modules over division rings, not just over fields. This is because much of the usual theory of vector spaces over fields still works over division rings (in particular, all modules are free). "Vector space" is not used more generally to refer to free modules over arbitrary rings.