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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 $\mathbb{H}^n$.

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

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  • $\begingroup$ could you give a example? I cannot imagine how two bases have different numbers of elements. Impossible for vector spaces and quaternionic spaces $\mathbb{H}^n$. $\endgroup$ – Shiquan Dec 11 '13 at 8:37
  • $\begingroup$ @RenShiquan There is an example in Hungerford that you can also find here. $\endgroup$ – user26857 Dec 11 '13 at 8:43

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