Characterisation of Linearly Independent 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 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:

$\map \Psi {\family {r_i}_{i \mathop \in I} } = 0$

if and only if:

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

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Thus injectivity and linearly independent are equivalent.

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$\blacksquare$