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Let $M$ be a module, finitely generated,over the principal ideal domain $D$. If $D$ is a field, then $M$ has a basis. But if $D$ is not a field theis is not truth.

Can you give me an example of a module over the integers $\mathbb Z$ finitely generated without a base?

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up vote 4 down vote accepted

Any finitely-generated torsion abelian group will do (and all such groups are actually finite). All elements are annihilated by some nonzero integer, and hence you have no linearly independent sets.

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Notice that torsion is a requirement here. Because finitely generated torsion-free modules over PIDs are free. – JSchlather Feb 23 '13 at 16:03

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