S-extending Fuzzy Modules

Hassan K.  Marhon

Hassan K. Marhon Ministry of Education, Resafa1

Keywords: Essential Fz-submodules, S-closed Fz-submodules, S-extending Fz-modules

Abstract

This paper introduces the S-extending fuzzy (fz) modules, i.e. the generalizations of classical semi extending modules in the setting of fuzzy modules. While much recent research on extending and FI-extending fz-modules has been productive, much less is known about how the new categories of CS-fz-modules, CLS-fz-modules, and FI-extending fz-modules are related to fz-modules. In this regard, we use a theoretical and proof based method to prove some necessary properties of S extending fz modules and relationships between these fuzzy module types. Our results show that each S-extending fz module verifies some conditions of direct summand decomposition, and thus it is worth consideration in the theory of modules. The results lay the foundation for further generalizations in algebraic structures, and do so with the possibility of applications in fuz set theory and computational mathematics.

References

H. K. Marhoon, Fuzzy Closed Submodules and Fuzzy W-closed Submodules with Some of Their Generalization, Ph.D. dissertation, Univ. of Baghdad, Iraq, 2020.

H. K. Marhon and H. Y. Khalaf, “Essential Fuzzy Submodules and Closed Fuzzy Submodules,” Iraq J. of Science, vol. 61, no. 4, pp. 890–897, 2020.

L. A. Zadeh, “Fuzzy Sets,” Information and Control, vol. 8, pp. 338–353, 1965.

M. M. Zahedi, “On L-Fuzzy Residual Quotient Module and P. Primary Submodule,” Fuzzy Sets and Systems, vol. 51, pp. 333–344, 1992.

L. Martinez, “Fuzzy Module Over Fuzzy Rings in Connection with Fuzzy Ideals of Rings,” J. Fuzzy Math., vol. 4, pp. 843–857, 1996.

Z. Yue, “Prime L-Fuzzy Ideals and Primary L-Fuzzy Ideals,” Fuzzy Sets and Systems, vol. 27, pp. 345–350, 1988.

R. Kumar, “Fuzzy Semi-Primary Ideals of Rings,” F-Sets and Systems, vol. 42, pp. 263–272, 1991.

H. J. Rabi, Prime Fuzzy Submodules and Prime Fuzzy Modules, M.Sc. thesis, Univ. of Baghdad, Iraq, 2001.

M. A. Hamil, Fuzzy Regular F-Modules, M.Sc. thesis, Univ. of Baghdad, Iraq, 2002.

M. A. Hamil, “Semi-Essential Fuzzy Ideals and Semi-Uniform Fuzzy Rings,” Ibn Al-Haitham J. for Pure and Appl. Sci., vol. 22, no. 4, 2009.

H. H. Abbas and Sh. N. Al-aeashi, “A Fuzzy Semi-Essential Submodule of A Fuzzy Module,” J. of Kufa for Math. and Comput., vol. 1, no. 5, pp. 31–37, 2012.

H. G. Rashed, Fully Cancellation Fuzzy Modules and Some Generalizations, M.Sc. thesis, Univ. of Baghdad, Iraq, 2017.

M. R. Abbas, St-Closed Submodules and Semi-Extending Modules, M.Sc. thesis, Univ. of Baghdad, Iraq, 2015.

M. A. Ahmed and Sh. O. Dakheen, “S-Maximal Submodules,” J. Baghdad for Science, vol. 12, no. 1, pp. 210–220, 2015.

H. K. Marhon and M. A. Hamel, “Some Properties of S-Closed Fuzzy Submodules,” J. of Iraqi Al-Khwarizmi (JIKH), vol. 7, pp. 205–211, 2023.

L. H. Rowen, Ring Theory, Academic Press Inc., 1991.

I. M. A. Hadi and M. A. Hamel, “Cancellation and Weakly Cancellation Fuzzy Modules,” J. of Basrah Researches, vol. 37, no. 4, 2011.

M. O. Behboodi, A. S. Karamzadeh, and H. Koohy, “Modules Whose Certain Submodules are Prime,” Vietnam Journal of Math., vol. 32, no. 3, pp. 303–317, 2004.

S. Abdulkadhim, S-Extending Modules and Related Concepts, Ph.D. dissertation, Al-Mustansiriya Univ., Iraq, 2007.

M. A. Hamel and H. Y. Khalaf, “Fuzzy Maximal Submodules,” Iraq J. of Science, vol. 61, no. 5, pp. 1164–1172, 2020.