<abstract><p>In this paper, we generalize a suitable transformation from an element-based to a submodule-based interpretation of the traditional idea of transitivity in QTAG modules. We examine QTAG modules that are transitive in the sense that the module has an automorphism that sends one isotype submodule $ K $ onto any other isotype submodule $ K' $, unless this is impossible because either the submodules or the quotient modules are not isomorphic. Additionally, the classes of strongly transitive and strongly $ U $-transitive QTAG modules are defined using a slight adaptations of this. This work investigates the latter class in depth, demonstrating that every $ \alpha $- module is strongly transitive with regard to countably generated isotype submodules.</p></abstract>

In this manuscript, we define the class of ω 1 -weakly α -projective QTAG-modules for the infinite ordinal α and provide its systematic study for the finite ordinal. Furthermore, we generalize this class to ω . 2 + n -projective modules and obtain some characterizations. We also study the ω -totally weak ω . 2 + n -projective modules under the formation of ω 1 -bijections.

This manuscript deals with the quasi-isomorphic invariants for QTAG modules; specially the cases when the module is summable, $$\sigma $$ σ -summable, $$(\omega +n)$$ ( ω + n ) -projective or HT -module. We show that if for a QTAG module M with a submodule N such that M/N is bounded, then M is weakly $$\omega _1$$ ω 1 -separable if and only if N is weakly $$\omega _1$$ ω 1 -separable.

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