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Note that if we have a ring $R$, we can talk about $R$-modules, and if we have a ring homomorphism $R\to S$, there is a map from $R$-modules to $S$-modules given by $-\otimes_RS$ (just assume everything is a bi-module, or adjust things or whatever). So is there some functor $(-)-$modules$:\underline{Rng}\to\mathcal{C}$ where $\mathcal{C}$ is some kind of category whose objects are categories of modules or something? Does this question even make sense??

Thanks! Jon

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    Sorry all, I've got a lot of unanswered questions out there, with answers more or less in the comments. I'd love for other people to turn these comments into answers, but it's sort of an ongoing project of mine to try to do this, at some point, but I worry it may not be until I finish with writing my thesis.2013-07-26

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