Lattice inside a finite dimensional vector space

Question

I have an integral domain $R$ and its field of fractions $K$. Let $V$ be a finite dimensional $K$ vector space. Let $M$ be a finitely generated $R$-module contained in $V$. Why is $K\cdot M=V$ equivalent to $M$ containing a $K$-basis of $V$?

Answer 1

Note that $KM$ is just the set of all $K$-linear combinations of elements of $M$. Of course, if $M$ contains a $K$-basis of $V$, then $KM = V$.

Conversely, if $KM = V$, say $M$ is generated by $v_1,\ldots,v_n$, so $KM = Kv_1 + \cdots + Kv_n$. Of course for $Kv_1 + \cdots + Kv_n$ to equal $V$ is to say that $v_1,\dots, v_n$ span $V$, so a subset of them form a basis for $V$. The $v_i$'s are in $M$, so $M$ contains a basis for $V$.