Definition:Basis of Module/Definition 2

Definition
Let $R$ be a ring with unity.

Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module. Let $\mathcal B = \family {b_i}_{i \mathop \in I}$ be a family of elements of $M$.

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

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

Also see

 * Equivalence of Definitions of Basis of Module