Characterisation of Linearly Independent Set through Free Module Indexed by Set

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Theorem

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

Let $S = \left\langle{m_i}\right\rangle_{i \mathop \in I}$ be a family of elements of $M$.

Let $\Psi : R^{\left({I}\right)} \to M$ be the 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:

$\Psi(\left\langle{r_i}\right\rangle_{i \mathop \in I}) = 0$

if and only if:

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



Thus injectivity and linearly independent are equivalent.



$\blacksquare$