Elements of a free module written uniquely as a linear combination of basis elements

Question

Let $R$ be a ring and $M$ a free $R$-module with basis $X$. Is it so that every $m \in M$ can be written uniquely as a linear combination of elements of $X$?

If not, in which cases is that true?

Update: In the book I'm learning from, a basis of $M$ is defined to be a linearly independent, spanning subset of $M$.

Answer 1

let $m = ∑r_i e_i = ∑s_i e_i$ then $∑(r_i-s_i) e_i=0$ so by independence we have $r_i=s_i$