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
Because a basis is a generator, this follows from Size of Spanning Set of Free Module is at Least Size of Basis.

Also see

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