Characterisation of Linearly Independent Set through Free Module Indexed by Set

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

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

Let $\Psi : R^{\paren I} \to M$ 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:
 * $\map \Psi {\family {r_i}_{i \mathop \in I} } = 0$


 * $\ds \sum_{i \mathop \in I} r_i m_i = 0$
 * $\ds \sum_{i \mathop \in I} r_i m_i = 0$

Thus injectivity and linearly independent are equivalent.