# Canonical Basis of Free Module on Set is Basis

## Theorem

Let $R$ be a ring with unity.

Let $I$ be a set.

Let $R^{(I)}$ be the free $R$-module on $I$.

Let $B$ be its canonical basis.

Then $B$ is a basis of $R^{(I)}$.

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Let $R$ be a ring with unity.

Let $I$ be a set.

Let $R^{(I)}$ be the free $R$-module on $I$.

Let $B$ be its canonical basis.

Then $B$ is a basis of $R^{(I)}$.

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