Bases of Free Module have Equal Cardinality

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Theorem

Let $R$ be a commutative ring with unity.

Let $M$ be a free $R$-module.

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


Then $B$ and $C$ are equivalent.

That is, they have the same cardinality.


Proof

By definition, a basis is a generator.

By Basis of Free Module is No Greater than Generator, there exist:

an injection $f : B \to C$
an injection $g : C \to B$

By the Cantor-Bernstein-Schröder Theorem, $B$ and $C$ are equivalent.

$\blacksquare$


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