We show the inheritance of summable property for certain fully invariant submodules by the QT AG-modules and vice versa. Important generalizations and extensions of classical results in this direction are also established.

A QTAG-module M is an α-module, where α is a limit ordinal, if M/H β (M ) is totally projective for every ordinal β < α. In the present paper α-modules are studied with the help of α-pure submodules, α-basic submodules, and α-large submodules. It is found that an α-closed α-module is an α-injective. For any ordinal ω ≤ α ≤ ω1 we prove that an α-large submodule L of an ω1-module M is summable if and only if M is summable. 2010 MSC: 16K20 R E T R A C T E D α-modules and generalized submodules 15 N = ∞ k=0 (N + H k (M )). Therefore the submodule N ⊆ M is closed with respect to h-topology if N = N . An h-reduced QTAG-module M is summable [14] if Soc(M ) = β<α S β , where S β is the set of all elements of H β (M ) which are not in H β+1 (M ), where α is the length of M . Moreover, M is called totally projective [10], if H α (Ext(M/H α (M ), M )) = 0 for all ordinal α and QTAG-modules M .It is interesting to note that almost all the results which hold for TAG-modules are also valid for QTAG-modules [13]. Our notations and terminology generally agree with those in [3] and [4].

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