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