# For every positive integer $i$, a free module $F$ with basis $B_1$ over a ring $R$ with identity has a basis $B_{i+1}$ with $|B_{i+1}|=|B_1|+i+1$.

Let $F$ be a free module over a ring $R$ with identity. Suppose $F$ has two finite basis $B_1,B_2$ such that $|B_2|=|B_1|+1$. Prove that for every positive integer $i$, $F$ has a basis $B_{i+1}$ with $|B_{i+1}|=|B_1|+i+1$. How to prove?

could you give a example? I cannot imagine how two bases have different numbers of elements. Impossible for vector spaces and quaternionic spaces $\mathbb{H}^n$.

• This is Exercise IV.2.12 in Hungerford, Algebra. – user26857 Dec 11 '13 at 8:20

If $R^n\simeq R^{n+1}$ then $R^{n+1}\simeq R^{n+2}$ (why?), so $R^n\simeq R^{n+2}$ and so on.
• could you give a example? I cannot imagine how two bases have different numbers of elements. Impossible for vector spaces and quaternionic spaces $\mathbb{H}^n$. – Shiquan Dec 11 '13 at 8:37