Let $R$ be a commutative ring and $M$ and $N$ be a free module module over $R$. The the rank(may infinite) of $M$ and $N$ are determined uniquely.
If there is a injection $M$ to $N$ ,then the cardinal of $rank(M)$ is equal to or less than the cardinal of $rank(N)$?
1,If $rank(N)$ is finite, I know the well-known proofs (using matrix or exterior algebra). 2,If $R$ is domain, it's so easy ( tensorig the quotient ring. If $R$ is a field ,the problem is just a problem for (finite-dim or infinite-dim) vector spaces)
I can't prove if $rank(N)$ is infinite.
(Sorry for my poor English)
I think "the proof " would use Zorn's lemma...
Or are there some counter-example?