Let $ F$ be a free $R$-module,with basis $w_1,..,w_n$. Then for $x \in F $ we have that $ x=w_1r_1+...+w_nr_n$ with $r_i \in R$ ,and $w_1,..,w_n$ are linearly independent over $R$. I have to prove that: $$F=w_1R \oplus ...\oplus w_nR$$
My idea is to prove that $w_iR \cap (w_1R+...+w_{i-1}R+w_{i+1}R...+w_nR) =0$ so the sum became direct sum.Is that the correct way and if it is how can i prove that?