A vector space is defined as a set $V$ together with a field $F$, such that $(V,+)$ is an abelian group, and $V$ has scalar multiplication that is compatible with $+$. But why is it that in this definition, we require $F$ to be a field, and not any commutative ring?
To illustrate, consider $V=\mathbb Z^n$, the set of all $n$ dimensional vectors with integer coordinates. Why doesn't it make sense to extend the definition of vector spaces such that we can say $V$ is a vector space over $\mathbb Z$? All the familiar properties remain: we certainly have closure under scalar multiplication, and the other axioms remain. In particular, the axioms say nothing about needing to have inverses, which is the only thing fields add on top of being a commutative ring. So why wouldn't ordinary commutative rings suffice?