Covers Induced by Ext

We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−, C)-cover, for any class of pure-injective modules C, and that (2) each module has a Ker Tor(−,B)-cover, for any class of left R-modules B. For Dedekind domains, we describe Ker Ext(−, C) explicitly for any class of cotorsion modules C; in particular, we prove that (1) holds, and that Ker Ext(−, C) is a cotilting torsion-free class. For right hereditary rings, we prove the consistency of the existence of special Ker Ext(−,G)-precovers for any set of modules G.