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A QTAG-module $M$ is an $\alpha$-module, where $\alpha$ is a limit ordinal, if $M/H_{\beta }\left(M\right)$ is totally projective for every ordinal $\beta <\alpha$. In the present paper $\alpha$-modules are studied with the help of $\alpha$-pure submodules, $\alpha$-basic submodules, and $\alpha$-large submodules. It is found that an $\alpha$-closed $\alpha$-module is an $\alpha$-injective. For any ordinal $\omega \le \alpha \le {\omega }_1$ we prove that an $\alpha$-large submodule $L$ of an ${\omega }_1$-module $M$ is summable if and only if $M$ is summable.