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 morphism given by Universal Property of Free Module Indexed by Set.

Then $S$ linearly independent iff $\Psi$ 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.