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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 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.
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)
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 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)