Some properties of Butler modules over valuation domains (Q1177664)

A torsion-free module \(B\) over a valuation domain \(R\) is a Butler module if for every torsion \(R\)-module \(T\) and for every exact sequence \(0\to T\to M\to B\to 0\) the induced map \(\text{Hom}(J/I,M)\to\text{Hom}(J/I,B)\) is surjective for all pairs \(I\subset J\) of \(R\)-submodules of the field of quotients of \(R\). The main result of the paper is: For any pure submodule \(N\) of rank \(k\) (=cardinal) of a reduced Butler module \(B\) and any rank one \(k'\)-generated module \(J\) with \(k'>k\), each epimorphism \(N\to J\) splits. This result leads to definitions of two classes of modules ( striped modules and modules having torsion extension property) which also are studied.