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Let $R$ be a ring and $R^n$ be the external sum of $n$ copies of the $R$-module $R$. Let's take $t$ elements of our $R$-module $R^n$ and call $M$ the sub-module generated by them. Then call $r$ the maximum number of independent elements among these $t$ elements. Taking such a subset $S$ of $t$ independent elements, it is not always true that $S$ spans $M$ (it is not always true that $M$ is free over $S$). If we take as $R$ a PID, then can such an unpleasant situations happen?

User
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    If I understand your question correctly, if one takes the ring $R=\mathbb{Z}, n=1$ and $S={2,3}$, the resulting module is $M=\mathbb{Z}$ abd one cannot choose a free basis of $M$ out of elements of $S$. So it is probably quite typical situation. – Pavel Čoupek Jun 08 '14 at 14:12
  • @PavelC Thanks! – User Jun 08 '14 at 15:14

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