Linear Algebra over a Ring/Free modules and matrices

By abuse of notation, we will write $$s$$ instead of $$f_s$$. Hence, the above proposition implies that we may denote an element $$m \in R\langle S \rangle$$ as a sum
 * $$m = \sum_{s \in S} a_s s$$,

where only finitely many $$a_s$$ are nonzero.