Definition:Basis of Module/Definition 2

Definition
Let $R$ be a ring with unity.

Let $\left({G, +_G, \circ}\right)_R$ 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 homomorphism given by Universal Property of Free Module on Set.

Then $\mathcal B$ is a basis $\Psi$ is an isomorphism.