Linearly independent subset of a module

Let \(R\) be a ring and \(M\) an \(R\)-module. A subset \(S\subseteq M\) is linearly independent if for all \(n\geq 1\), all pairwise distinct \(s_1, \ldots, s_n\in S\) and all \(r_1, \ldots, r_n\in R\), \[ r_1 s_1 + \cdots + r_n s_n = 0 \implies r_1 = \cdots = r_n = 0 .\]