Cotorsion theories and splitters

Abstract

Let R R be a subring of the rationals. We want to investigate self splitting R R -modules G G (that is Ext R ⁡ ( G , G ) = 0 ) \operatorname {Ext}_R(G,G) = 0) . Following Schultz, we call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples of splitters. Are there others? Answering an open problem posed by Schultz, we will show that there are more splitters, in fact we are able to prescribe their endomorphism R R -algebras with a free R R -module structure. As a by-product we are able to solve a problem of Salce, showing that all rational cotorsion theories have enough injectives and enough projectives. This is also basic for answering the flat-cover-conjecture.