Every set with more than $k$ elements is dependent 
If $F$ is a free $R-$module, $R$ a domain, with rank=$k$ and $X\subset F$ show that $X$ is linear independent $\Rightarrow |X|\leq k$

There is a hint saying "consider the embedding of $R$ in it's field of fractions $R'$"
Attempt
Let $\phi:R\to R'$ be this embedding and $X\subset F$ with $|X|>k,X=\{x_1,...,x_m\}$
We may consider $F$ as a $R'-$module ($*$) so $R'-$ vector space and then since $dim_{R'}F=k$, $X$ is linear dependent over $R'$. Then I showed that $X$ is linear dependent over $R$.
So my question is :
How can I justify $(*)$ (if it is true)?? Is there a quicker way to answer this exercise??
Thanks!

A priori you can only view F as an R-module.
However,
the span of the image under a linear extension of the embedding can be viewed as a free module over the field of fractions. I.e. as a vector space over the field of fractions.
The nice thing is, that the embedding preserves the property of being linearly independent over R. Now the only thing you need to prove is that linear independence over R implies linear independence over R’  when R is a domain (multiply by every denominator in the R’ linear combination).
I hope this helps.
