# Thread: free R-module, subset basis

1. ## free R-module, subset basis

Let $R$ be a ring and $M$ a free $R-$module. Prove that a subset $S \subseteq M$ is a basis for $M$ if and only if every $m \in M$ may be expressed uniquely as a linear combination of elements of $S.$

2. S generates M <=> existence

S is free <=> unicity