I have recently been doing some calculations in topology which are naturally expressed in terms commutative ring objects in the category of cocommutative coalgebras. These have been studied for quite a while - initially by Milgram and Ravenel-Wilson since they arise in studying homology of ring spectra - and go by the name "Hopf rings" (for better or worse).
I'd like to make the calculations more "meaningful" if possible so my question is: how much of commutative algebra has been reproduced for ring objects in categories other than
vector spaces sets? For some suitably nice categories C, hopefully including cocommutative coalgebras, do ring objects in C have analogues of ideals/modules, dimension, Spec, localization,...? I realize this is open-ended, but if there is existing work along any of these lines it would be nice to look and see if the ring objects I am considering fit in with and are illuminated by such a framework. Since (an interesting subset of) cocommutative Hopf algebras are given by group objects in the category of cocommutative coalgebras, one would suspect that this category's ring objects are also particularly nice but I am not aware of any such general development.