Span of a f.g. $R$-module over the quotient field of $R$.

Question

Let $R$ be an integral domain with quotient field $K$, $R \neq K$. Let $V$ be a finite dimensional vector space over $K$ and $M$ a finitely generated $R$-submodule of $V$. My question is how $KM=V$ is equivalent to $M$ containing a $K$-basis of $V$?

Answer 1

I'm assuming you mean $M$ to be a submodule of $V$. By definition, $KM$ is the set of $K$-linear combinations of vectors in $M$. This is a $K$-subspace of $V$. In fact, since $M$ itself is finitely generated over $R$, the set $KM$ is equal to the set of $K$-linear combinations of a set of generators $x_1,\ldots,x_n\in M$ for $M$ as an $R$-module. In general, $KM$ is therefore a $K$-subspace of $V$, and it is spanned by the $x_i$. So, if $KM=V$, then the $x_i$ span $V$ over $K$, and therefore some subset of the $x_i$ must constitute a $K$-basis for $V$. Conversely, if there is a $K$-basis for $V$ contained in $M$, then certainly the span of $M$ over $K$ contains the span of the basis, which is $V$, so $KM=V$.