The concept of a QTAG-module M R was given by Singh [8]. The structure theory of such modules has been developed on similar lines as that of torsion abelian groups. If a module M R is such that M ⊕ M is a QTAG-module, it is called a strongly TAG-module. This in turn leads to the concept of a primary TAG-module and its periodicity. In the present paper some decomposition theorems for those primary TAG-modules in which all h -neat submodules are h -pure are proved. Unlike torsion abelian groups, there exist primary TAG-modules of infinite periodicities. Such modules are studied in the last section. The results proved in this paper indicate that the structure theory of primary TAG-modules of infinite periodicity is not very similar to that oftorsion abelian groups.