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