So I'm stuck with this problem.
Let $A$ be a nonzero commutative ring (with unit). I have several questions that are really close to each other.
1) Let $M$ be a finitely generated module over $A$. Can we have $\{x_n, n \in \mathbb{N}\} \subset M$ a linear independent family in $M$ ?
2) Let $M$ be a module over $A$ generated by $f_1, \ldots, f_n \in M$. Can we have $x_1,\ldots, x_{n+1} \in M$ a linear independent family ? (Obviously a negative answer to the latter implies a negative answer to the former.)
3) Let $N$ be a free module over $A$, and $M$ a module such that there exists an exact sequence $0 \to N \to M$ and another exact sequence $N \to M \to 0$. Is $M$ a free module as well (is $M \simeq N$)? If not, can we add $N$ of finite rank to get a positive answer ?
4) Let $M$ be finitely generated by $f_1, \ldots, f_n$, and $x_1, \ldots, x_n$ be a linear independent family in $M$. Is $M$ free?
Thank you for your help!