\(h\)-purifiable submodules and isomorphism of \(h\)-pure hulls (Q345212)

Summary: Let \(N\) be a \(h\)-purifiable submodule of a QTAG-module \(M\) and \(K\) be a \(h\)-pure hull of \(N\) in \(M\). Then \(K\) is a direct summand of \(M\) if and only if \(\mathrm{Soc}(M)/\mathrm{Soc}(N)\) is \(h\)-purifiable in \(M/\mathrm{Soc}(N)\). Also, if \(K\) is a direct summand of \(M\), then all \(h\)-pure hulls of \(N\) are direct summands of \(M\), there exists the same complementary summand of \(M\) for every \(h\)-pure hull of \(N\), and all \(h\)-pure hulls of \(N\) are isomorphic.