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first question on stack exchange, please let me know if I have made any errors with formatting or in general! :)

Let $f_1,f_2, ...,f_s$ be a basis of a free module $V$ over a PID $R$. Suppose that $f=r_1f_1+r_2f_2+... + r_sf_s$ and that $1$ is a gcd of {$r_1,r_2,...,r_s$}. Show that $f$ is a part of a basis for $V$.

My first thought was that any element in a free module over a PID could be an element of some basis, but then realised that $\mathbb{Z}$ (which is a PID) is a free module over itself and any non-unit is not a basis element. (My suspicion is existence of non-trivial ideals in a PID means not every element is necessarily part of a basis as some elements do not generate the whole set). I guess this is something that is true for vector spaces but not for modules.

I feel as though if I can find some sort of automorphism where one of the basis elements is sent to $f$ I would have it, however I've tried to think of a map and they have tended to be pretty unintuitive and messy so far. Any help would be appreciated, thanks!

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  • $\begingroup$ When faced with any such problem, it is always wise to try `small' cases first. For $s=1$, I hope the assertion is obvious. Can you do this for $s=2$? Does this suggest what you could do for general $s$? $\endgroup$ – Mohan May 9 '16 at 14:31

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