Characterisation of Spanning Set through Free Module Indexed by Set

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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 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 $\family {r_i}_{i \mathop \in I} \in R^{\paren I}$ we have:

$\map \Psi {\family {r_i}_{i \mathop \in I} } = \ds \sum_{i \mathop \in I} m_i r_i$

Thus $\Psi$ is surjective if and only if every element of $M$ is a linear combination of $S$.

$\blacksquare$