Let $R$ be a commutative ring with the identity and $F$ be a free-module over $R$.
I assume $X$ is the basis of $F$.I know any two bases of $F$ have the same cardinality.
My question is that can we get that the cardinality of the maximal linearly independent susbet of F is less than the cardinality of $X$, ie if $Y$ is any maximal linearly independent susbet of F,can we get $|Y|\le |X|$.
If $|X| $ is finite, it is true. How about $|X|$ is infinite?
Thanks!