# Free module over a ring with identity with a basis of size $m, \ \forall m \ge n$

Please, help on this Exercise [Hungerford's Algebra, IV.2.12]

If $F$ is a free module over a ring with identity such that $F$ has a basis of finite cardinality $n\geq 1$ and another basis of cardinality $n+1$, then $F$ has a basis of cardinality $m$ for every $m\geq n$.

I suposse that I need to use induction but...

Let $R$ be ring for which $F$ is a free module. Then we have isomorphisms $R^n \rightarrow F$ and $F \rightarrow R^{n+1}$ which gives us an isomorphism $R^n \rightarrow R^{n+1}$. Thus for any $m \geq n$ we have an isomorphism $R^m \cong R^n \oplus R^{m-n} \cong R^{n+1} \oplus R^{m-n} \cong R^{m+1}$. Composing these isomorphisms we get an isomorphism $R^n \cong R^m$ for all $m \geq n$. Now use the isomorphism $R^n \cong F$ to find that $F \cong R^m$, that is, $F$ has a basis consisting of $m$ elements.