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 present Shelah's famous theorem in a version for modules, together with a self-contained proof and some examples. This exposition is based on lectures given at CRM in October 2006. The Singular Compactness Theorem is about an abstract notion of " free ". The general form of the theorem is as follows: If λ is a singular cardinal and M is a λ-generated… (More)
We describe a general construction of a module A from a given module B such that Ext(B, A) = 0 and we apply it to answer several questions on splitters, cotorsion theories, and saturated rings.
Let R be a ring and T be a 1-tilting right R-module. Then T is of countable type. Moreover, T is of finite type in case R is a Prüfer domain.
We show that the concept of an Abstract Elementary Class (AEC) provides a unifying notion for several properties of classes of modules and discuss the stability class of these AEC. An abstract elementary class consists of a class of models K and a strengthening of the notion of submodel ≺ K such that (K, ≺ K) satisfies the properties described below. Here… (More)
We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecom-posable, that is, they remain indecomposable in any generic extension. However, any absolutely rigid family of groups has cardinality less than the partition cardinal κ(ω). Added December 2004: The proofs of Theorems… (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)