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Is it possible to use the ring $R = \mathbb{Z}/4\mathbb{Z}$ to construct a counter-example that submodules of free modules are not necessarily free?

Thanks a lot.

2 Answers 2

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    A free module is a module isomorphic to $R^n$, this latter has $4^n$ elements since $R$ has four of them.2012-03-31
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For any commutative ring $R$ with unity, $R$ is a free module over itself. If $0\ne a\in R$ is a zero-divisor, then the principal ideal generated by $a$ as a submodule of $R$ is not free. So your example works.