Free Module is Isomorphic to Free Module on Set

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Theorem

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

Let $\mathcal B = \left\langle{b_i}\right\rangle_{i \mathop \in I}$ 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 on Set.


Then the following are equivalent:

$\mathcal B$ is a basis of $M$
$\Psi$ is an isomorphism


Proof

Follows directly from:

Characterisation of Linearly Independent Set through Free Module Indexed by Set
Characterisation of Spanning Set through Free Module Indexed by Set.

$\blacksquare$