Some functors arising from the consideration of torsion theories over noncommutative rings

To each associative (but not necessarily commutative) ring R we assign the complete distributive lattice R-tors of (hereditary) torsion theories over R-mod. We consider two ways of making this process functorial – once contravariantly and once covariantly – by selecting appropriate subcategories of the category of associative rings. Combined with a functor due to Rota, this gives us functors from these subcategories to the category of commutative rings.