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Retracted: According to the retraction notice https://doi.org/10.2478/cm-2019-0014, this paper has been retracted due to a large coincidence with [P. V. Danchev, SUT J. Math. 44, No. 1, 33–37 (2008; Zbl 1158.20325)].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.
$\alpha$-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.
References:
[1] Ansari, A. H., Ahmad, M., Khan, M.Z.: Some decomposition theorems on $S_2$-modules. III. Tamkang J. Math., 12, 2, 1981, 147-154, MR 0676096
[2] Benabdallah, K., Singh, S.: On torsion abelian groups like modules. Lecture Notes in Mathematics, Springer Verlag, 1006, 1983, 639-653, MR 0722656
[3] Fuchs, L.: Infinite Abelian Groups. 1970, Academic Press, New York, Vol. I. MR 0255673
[4] Fuchs, L.: Infinite Abelian Groups. 1973, Academic Press, New York, Vol. II. MR 0349869 | Zbl 0257.20035
[5] Hasan, A.: On essentially finitely indecomposable QTAG-modules. Afrika Mat., 27, 1, 2016, 79-85, DOI 10.1007/s13370-015-0318-7 | MR 3459317
[6] Hasan, A.: On generalized submodules of QTAG-modules. Georgian Math. J., 23, 2, 2016, 221-226, DOI 10.1515/gmj-2015-0061 | MR 3507951
[7] Hasan, A., Rafiquddin: Notes on summability in QTAG-modules. Tbilisi Math. J., 10, 2, 2017, 235-242, DOI 10.1515/tmj-2017-0039 | MR 3689617
[8] Hasan, A., Rafiquddin, Ahmad, M.F.: On $\alpha $-modules and their applications. Southeast Asian Bull. Math., 2019, To appear. MR 3966293
[9] Khan, M.Z.: $h$-divisible and basic submodules. Tamkang J. Math., 10, 2, 1979, 197-203, MR 0584534
[10] Mehdi, A., Abbasi, M.Y., Mehdi, F.: Nice decomposition series and rich modules. South East Asian J. Math. & Math. Sci., 4, 1, 2005, 1-6, MR 2208767
[11] Mehdi, A., Naji, S.A.R.K., Hasan, A.: Small homomorphisms and large submodules of QTAG-modules. Sci. Ser. A. Math Sci., 23, 2012, 19-24, MR 2961700
[12] Mehdi, A., Sikander, F., Naji, S.A.R.K.: Generalizations of basic and large submodules of QTAG-modules. Afrika Mat., 25, 4, 2014, 975-986, DOI 10.1007/s13370-013-0167-1 | MR 3277863
[13] Mehran, H., Singh, S.: On $\sigma $-pure submodules of QTAG-modules. Arch. Math., 46, 1986, 501-510, DOI 10.1007/BF01195018 | MR 0849855
[14] Naji, S.A.R.K.: A study of different structures in QTAG-modules. 2010, Ph.D. Thesis, Aligarh Muslim University.
[15] Singh, S.: Some decomposition theorems in abelian groups and their generalizations. Ring Theory: Proceedings of Ohio University Conference 25, 1976, 183-189, Marcel Dekker, New York, MR 0435146
[16] Singh, S.: Abelian groups like modules. Act. Math. Hung, 50, 1987, 85-95, DOI 10.1007/BF01903367 | MR 0893248
[17] Singh, S., Khan, M.Z.: TAG-modules with complement submodules $h$-pure. Internat. J. Math. & Math. Sci., 21, 4, 1998, 801-814, DOI 10.1155/S0161171298001112 | MR 1642265
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