\begin{definition}[Definition:Basis of Module/Definition 2]
Let $R$ be a ring with unity.
Let $\struct {G, +_G, \circ}_R$ 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 homomorphism given by Universal Property of Free Module on Set.
Then $\BB$ is a '''basis''' {{Iff}} $\Psi$ is an isomorphism.
\end{definition}