Characterisation of Spanning 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 $S$ is a spanning set of $M$ iff $\Psi$ is surjective.

Proof
For $(r_i)_{i\in I}\in R^{(I)}$ we have
 * $\Psi((r_i)_{i\in I})=\sum_{i\in I}m_ir_i$

Thus $\Psi$ is surjective iff every element of $M$ is a linear combination of $S$.