Subsocles and direct sum of uniserial modules

Yıl 2021, Cilt: 8 Sayı: 3, 213 - 218, 15.09.2021

Ayazul Hasan

Öz

Suppose $M$ is a $QTAG$-module with a subsocle $S$ such that $M/S$ is a direct sum of uniserial modules. Our global aim here is to investigate an interesting connection between the structure of $M/S$ and the $QTAG$-module $M$. Specifically, the condition $S=Soc(N)$ for some $h$-pure submodules $N$ of $M$ allows $M$ to inherit the structure of $M/S$.

Anahtar Kelimeler

QTAG-modules , $h$-pure submodules , Socles

Kaynakça

• [1] M. Ahmad, A. H. Ansari, M. Z. Khan, On subsocles of S2-modules, Tamkang J. Math. 11(2) (1980) 221-229.
• [2] A. Facchini, L. Salce, Uniserial modules: sums and isomorphisms of subquotients, Comm. Algebra 18(2) (1990) 499-517.
• [3] L. Fuchs, Infinite Abelian groups, Volume I, Pure Appl. Math. 36, Academic Press, New York (1970).
• [4] L. Fuchs, Infinite Abelian groups, Volume II, Pure Appl. Math. 36, Academic Press, New York (1973).
• [5] A. Hasan, On essentially finitely indecomposable QTAG-modules, Afr. Mat. 27(1) (2016) 79-85.
• [6] A. Hasan, Rafiquddin, On completeness in QTAG-modules, (Communicated).
• [7] J. Irwin, J. Swanek, On purifiable subsocles of a primary Abelian group, Can. J. Math. 23(1) (1971) 48-57.
• [8] M. Z. Khan, Torsion modules behaving like torsion Abelian groups, Tamkang J. Math. 9(1) (1978) 15-20.
• [9] M. Z. Khan, Modules behaving like torsion Abelian groups, Math. Japonica, 22(5) (1978) 513-518.
• [10] A. Mehdi, On some QTAG-modules, Sci. Ser. A. Math Sci. 21 (2011) 55-62.
• [11] 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.
• [12] A. Mehdi, S. A. R. K. Naji, A. Hasan, Small homomorphisms and large submodules of QTAGmodules, Sci. Ser. A. Math Sci. 23 (2012) 19–24.
• [13] A. Mehdi, F. Sikander, S. A. R. K. Naji, Generalizations of basic and large submodules of QTAGmodules, Afr. Mat. 25(4) (2014) 975-986.
• [14] H. A. Mehran, S. Singh, On -pure submodules of QTAG-modules, Arch. Math. 46(6) (1986) 501–510.
• [15] F. Sikander, A. Hasan, A. Mehdi, On n-layered QTAG-modules, Bull. Math. Sci. 4(2) (2014) 199-208.
• [16] S. Singh, Some decomposition theorems in Abelian groups and their generalizations, Ring Theory: Proceedings of Ohio University Conference, Marcel Dekker, New York 25 (1976) 183-189.
• [17] S. Singh, Abelian groups like modules, Act. Math. Hung, 50 (1987) 85-95.