Geodesic
$\alpha$-modules and generalized submodules
Communications in Mathematics, Tome 27 (2019) no. 1, pp. 13-26
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A QTAG-module $M$ is an $\alpha$-module, where $\alpha $ is a limit ordinal, if $M/H_\beta (M)$ 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 $\omega \leq \alpha \leq \omega _1$ we prove that an $\alpha $-large submodule $L$ of an $\omega _1$-module $M$ is summable if and only if $M$ is summable.
A QTAG-module $M$ is an $\alpha$-module, where $\alpha $ is a limit ordinal, if $M/H_\beta (M)$ 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 $\omega \leq \alpha \leq \omega _1$ we prove that an $\alpha $-large submodule $L$ of an $\omega _1$-module $M$ is summable if and only if $M$ is summable.
Classification : 16K20
Keywords: $\alpha$-modules; $\alpha$-pure submodules; $\alpha$-basic submodules; $\alpha$-large submodules. 
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Rafiquddin, Rafiquddin; Hasan, Ayazul; Ahmad, Mohammad Fareed. $\alpha$-modules and generalized submodules. Communications in Mathematics, Tome 27 (2019) no. 1, pp. 13-26. http://geodesic.mathdoc.fr/item/COMIM_2019_27_1_a1/

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