On submodule transitivity of QTAG-modules

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 $K^{\prime }$, 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.