Free Module is Isomorphic to Free Module on Set

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.