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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(More)
We consider the Whitehead problem for principal ideal domains of large size. It is proved, in ZFC, that some p.i.d.'s of size ≥ ℵ 2 have non-free Whitehead modules even though they are not complete discrete valuation rings. A module M is a Whitehead module if Ext 1 R (M, R) = 0. The second author proved that the problem of whether every Whitehead Z-module(More)
Introduction. This survey is intended to introduce to logicians some notions , methods and theorems in set theory which arose—largely through the work of Saharon Shelah—out of (successful) attempts to solve problems in abelian group theory, principally the Whitehead problem and the closely related problem of the existence of almost free abelian groups.(More)