About nonabelian modules over localizations of Z, I knew all that and was wondering about non-torsion cases.

]]>Anyway, remember rings are monoids internal to Ab, and modules are actions internal to Ab. The immediate problem with replacing Ab with Grp is that there's no reasonable way to put a closed monoidal category structure on Grp which respects the underlying set functor, because Hom_Grp(A,B) has no reasonable group structure. I think what's going on with near-rings and their modules is you give a (nonsymmetric) closed monoidal structure whose internal hom is Hom_Set(A,B) with pointwise group operation, leading to a nonsymmetric notion of left/right near-rings and right modules over right near-rings / left modules over left near-rings.

As far as localizations of the integers go, any localization is just a specified collection of invertible primes, and the torsion modules will just be ones where the exponent doesn't have one of those prime factors. In particular p-groups would just be torsion Z_{(p)}-modules, and no torsion groups are Q-modules.

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