# Definition:Basis of Module/Definition 2

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## Definition

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 of $G$** if and only if $\Psi$ is an isomorphism.

## Also see

## Linguistic Note

The plural of **basis** is **bases**.

This is properly pronounced **bay-seez**, not **bay-siz**.

## Sources

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