# Bases of Finitely Generated Free Module have Equal Cardinality

## Theorem

Let $R$ be a commutative ring with unity.

Let $M$ be finitely generated.

Let $B$ and $C$ be bases of $M$.

Then $B$ and $C$ are finite and have the same cardinality.

## Proof

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Let $R$ be a commutative ring with unity.

Let $M$ be finitely generated.

Let $B$ and $C$ be bases of $M$.

Then $B$ and $C$ are finite and have the same cardinality.

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