Suppose $M$ is a n-dimensional free module over a commutative ring with identity , is the number of elements in every linearly independent set less than or equal to n?
I have prooven it is right when R is an integral domain. Because the ring has fractional field, then we can proof that the columns of a matrix $A_{n \times m}$ is linearly dependent when $n＜ m$.
This is a classic question: https://mathoverflow.net/q/136/157421
I really like the answer by Robin: http://mathoverflow.net/questions/30860/ranks-of-free-submodules-of-free-modules/30862#30862. His proof shows that the columns of a matrix $A_{n \times m}$ over a commutative ring is linearly dependent when $n ＜m$. The most important is that this proof doesn't need other backgroud except pure matrix theory.
P.S: I also construct an example about a linearly independent element in a 2-dimensional free module over an integral domain can not be expanded to a basis.
