Bases of Finitely Generated Free Module have Equal Cardinality

Theorem
Let $R$ be a commutative ring with unity.

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

Let $M$ be finitely generated.

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

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

Also see

 * Bases of Finitely Generated Vector Space have Equal Cardinality