Bases of Free Module have Equal Cardinality
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Theorem
Let $R$ be a commutative ring with unity.
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:
By the Cantor-Bernstein-Schröder Theorem, $B$ and $C$ are equivalent.
$\blacksquare$