Cotorsion Theories Cogenerated by $\aleph_1$-free Abelian Groups

Cotorsion theories cogenerated by א1-free abelian groups

Given an א1-free abelian group G we characterize the class CG of all torsion abelian groups T satisfying Ext(G, T ) = 0 assuming the special continuum hypothesis CH. Moreover, in Gödel’s constructable universe we prove that this characterizes CG for arbitrary torsion-free abelian G. It follows that there exist some ugly א1-free abelian groups.

Cotorsion Theories and Splitters

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is ExtR(G,G) = 0 holds and follow Schultz [22] to call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples for splitters. Are there others? Answering an open problem by Schultz [22] we will show that there are more splitters, in fact we are able to prescribe the...

Intuitionistic Free Abelian Groups

Strong theories of ordered abelian groups

We consider strong expansions of the theory of ordered abelian groups. We show that the assumption of strength has a multitude of desirable consequences for the structure of definable sets in such theories, in particular as relates to definable infinite discrete sets. We also provide a range of examples of strong expansions of ordered abelian groups which demonstrate the great variety of such t...

Undecidable theories of valuated abelian groups

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