| context | $V,F$ |
| definiendum | $\langle\mathcal V,\mathcal F, *\rangle \in \mathrm{vectorSpace}(V,F)$ |
| context | $\langle\mathcal V,\mathcal F, *\rangle \in \mathrm{module}(V,F)$ |
| context | $\mathcal F\in \mathrm{field}(F)$ |
A vector space is a $F$-module over $V$, where $F$ is a field, not just a ring.
One speaks of an $F$-vector space over $V$. Here $F$ and $V$ are just sets.
Wikipedia: Vector space