Characterisation of Linearly Independent Set through Free Module Indexed by Set

Theorem
Let $M$ be a unitary $R$-module.

Let $S = (m_i)_{i\in I}$ be a family of elements of $M$.

Let $\Psi : R^{(I)} \to M$ be the module homomorphism given by Universal Property of Free Module on Set.

Then the following are equivalent:
 * $S$ linearly independent
 * $\Psi$ is injective.

Proof
We have $\Psi((r_i)_{i\in I}) = 0$ iff $\sum_{i\in I}r_im_i=0$.

Thus injectivity and linear independence are equivalent.