Talk:Finitely-generated module
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A modification of the previous example gives a finitely-generated module: take the rational numbers whose denominator is strictly less than 6 (after simplification: 16/12=4/3, so it belongs to this set). This is a module over the integers, which is also finitely generated. A set of generators is, for example, {1/1,1/2,1/3,1/4,1/5}, but also {1/3,1/4,1/5} (this one is minimal).
This isn't a module. Since 1/4 is cited as part of a minimal generating set, I assume the intention is that addition in the module is addition of rationals, and multiplication by a coefficient is ordinary multiplication.
In that case, observe that 1/4 + (-1)*(1/5) = 1/20 is not in the set described. So the set described isn't closed and hence is not a module.
Also, in "modifying the previous example" the meanings of addition and multiplication in the module have changed. This should be explicitly stated.
SORRY!!!!!!! Sorry. Sorry. Sorry. I have been misguided for my constant thinking on semigroups and things with a common denominator. You are absolutely right. I am going to re-edit it right now. Pfortuny
What I wanted to do was showing a simple example of a f.g. module for which any set of generators is linearly dependent. This is trivial with torsion modules, but I did not want to mess up a casual reader. Any ideas? Pfortuny
