Bases of Free Module have Equal Cardinality

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 equinumerous.

Proof
By definition, a basis is a generator.

From Size of Spanning Set of Free Module is at Least Size of Basis, 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.

Also see

 * Bases of Finitely Generated Free Module have Equal Cardinality
 * Bases of Vector Space have Equal Cardinality