I am currently reading Dummit & Foote and i get confused when encountered with two definitions of free modules. One which i read on wikipedia says that "A free module is a module with a basis"
The second one which i read in book says " An R module F is said to be free on a subset A of F if for all $x\neq 0$ there exists unique nonzero elements $r_{1},r_{2}, ...r_{n}$ of R and unique $a_{1},a_{2}, ...a_{n}$ in A such that $x=r_{1}a_{1}+...+r_{n}a_{n}$ for some n in positive integers.
How can we say two definitions are equivalent? For second definition imply one we need to show for any finite subset $a_{1},a_{2},...,a_{n}$ of A, $r_{1}a_{1}+...+r_{n}a_{n}=0$ implies each $r_{i}$ is zero. But second definition only talk about nonzero elements of F. Please provide me a hint or tell me is there anything wrong assumed in definition second?
The second definition $\rightarrow$ The first definition.
We have to show that $F$ has a basis $A$, i.e. for any finite subset $a_1,a_2,...,a_n$ of $A$, $r_1a_1+...+r_na_n=0$ implies each $r_i$ is zero. Suppose there is some $r_i \not=0$ (without loss of generality, $r_1 \not =0$), then $$r_1a_1=-r_2a_2- \cdots -r_na_n,$$ which means the element $r_1a_1$ has two expressions, contradict to the second definition. Hence $r_1=0$. Similarly, each $r_i$ is zero.
The first defintion $\rightarrow$ The second definition
I think you can prove it.