Characterisation of Spanning Set through Free Module Indexed by Set
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Theorem
Let $M$ be a unitary $R$-module.
Let $S = \left\langle{m_i}\right\rangle_{i \mathop \in I}$ be a family of elements of $M$.
Let $\Psi: R^{\left({I}\right)} \to M$ be the morphism given by Universal Property of Free Module Indexed by Set.
Then $S$ is a spanning set of $M$ if and only if $\Psi$ is surjective.
Proof
For $\left\langle{r_i}\right\rangle_{i \mathop \in I} \in R^{\left({I}\right)}$ we have:
- $\Psi \left({\left\langle{r_i}\right\rangle_{i \mathop \in I}}\right) = \displaystyle \sum_{i \mathop \in I} m_i r_i$
Thus $\Psi$ is surjective if and only if every element of $M$ is a linear combination of $S$.
$\blacksquare$