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torsion theory; torsionfree module; $\sigma$-QTAG-module; kernel of purity; center of purity
The structure theory of abelian $p$-groups does not depend on the properties of the ring of integers, in general. The substantial portion of this theory is based on the fact that a finitely generated $p$-group is a direct sum of cyclics. Given a hereditary torsion theory on the category $R$-{\bf Mod} of unitary left $R$-modules we can investigate torsionfree modules having the corresponding property for all torsionfree factor-modules (and a natural requirement concerning extensions of some homomorphisms). This paper continues in our previous investigations of the structural properties of such modules.
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