Free Module is Isomorphic to Free Module on Set

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Theorem

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

Let $\BB = \family {b_i}_{i \mathop \in I}$ be a family of elements of $M$.

Let $\Psi: R^{\paren I} \to M$ be the morphism given by Universal Property of Free Module on Set.

Then the following are equivalent:

$\BB$ 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.

$\blacksquare$