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Article

Rafiquddin, Rafiquddin ; Hasan, Ayazul ; Ahmad, Mohammad Fareed
$\alpha$-modules and generalized submodules. (English). Communications in Mathematics, vol. 27 (2019), issue 1, pp. 13-26
MSC: 16K20 | MR 3977474 | Zbl 1464.16013
Retracted: According to the retraction notice https://doi.org/10.2478/cm-2019-0014, this paper has been retracted due to a large coincidence with [P. V. Danchev, SUT J. Math. 44, No. 1, 33–37 (2008; Zbl 1158.20325)].
Keywords:
$\alpha$-modules; $\alpha$-pure submodules; $\alpha$-basic submodules; $\alpha$-large submodules.
Summary:
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.
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