α-modules and generalized submodules
Abstract
A QTAG-module M is an α-module, where α is a limit ordinal, if M/Hβ (M) is totally projective for every ordinal β < α. In the present paper α-modules are studied with the help of α-pure submodules, α-basic submodules, and α-large submodules. It is found that an α-closed α-module is an α-injective. For any ordinal ω ≤ α ≤ ω1 we prove that an α-large submodule L of an ω1-module M is summable if and only if M is summable.
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[1] A. H. Ansari, M. Ahmad, M.Z. Khan: Some decomposition theorems on S2-modules. III. Tamkang J. Math. 12 (2) (1981) 147–154.
-
[2] K. Benabdallah, S. Singh: On torsion abelian groups like modules. Lecture Notes in Mathematics, Springer Verlag 1006 (1983) 639–653.
-
[3] L. Fuchs: Infinite Abelian Groups. Academic Press, New York (1970). Vol. I
-
[4] L. Fuchs: Infinite Abelian Groups. Academic Press, New York (1973). Vol. II
-
[5] A. Hasan: On essentially finitely indecomposable QTAG-modules. Afrika Mat. 27 (1) (2016) 79–85.
-
[6] A. Hasan: On generalized submodules of QTAG-modules. Georgian Math. J. 23 (2) (2016) 221–226.
-
[7] A. Hasan, Rafiquddin: Notes on summability in QTAG-modules. Tbilisi Math. J. 10 (2) (2017) 235–242.
-
[8] A. Hasan, Rafiquddin, M.F. Ahmad: On α-modules and their applications. Southeast Asian Bull. Math. (2019). To appear
-
[9] M.Z. Khan: h-divisible and basic submodules. Tamkang J. Math. 10 (2) (1979) 197–203.
-
[10] A. Mehdi, M.Y. Abbasi, F. Mehdi: Nice decomposition series and rich modules. South East Asian J. Math. & Math. Sci. 4 (1) (2005) 1–6.
-
[11] A. Mehdi, S.A.R.K. Naji, A. Hasan: Small homomorphisms and large submodules of QTAG-modules. Sci. Ser. A. Math Sci. 23 (2012) 19–24.
-
[12] A. Mehdi, F. Sikander, S.A.R.K. Naji: Generalizations of basic and large submodules of QTAG-modules. Afrika Mat. 25 (4) (2014) 975–986.
-
[13] H. Mehran, S. Singh: On σ-pure submodules of QTAG-modules. Arch. Math. 46 (1986) 501–510.
-
[14] S.A.R.K. Naji: A study of di erent structures in QTAG-modules. Ph.D. Thesis, Aligarh Muslim University (2010)
-
[15] S. Singh: Some decomposition theorems in abelian groups and their generalizations. In: Ring Theory: Proceedings of Ohio University Conference 25. Marcel Dekker, New York (1976) 183–189.
-
[16] S. Singh: Abelian groups like modules. Act. Math. Hung 50 (1987) 85–95.
-
[17] S. Singh, M.Z. Khan: TAG-modules with complement submodules h-pure. Internat. J. Math. & Math. Sci. 21 (4) (1998) 801–814.
