jaem TWMS Journal of Applied and Engineering Mathematics 2146-1147 2587-1013 Işık University Press Algebra and Number Theory Cebir ve Sayı Teorisi A STUDY OF FUZZY $\mathcal{W}$-CLOSED SUBMODULES AND RELATED CONCEPTS https://orcid.org/0000-0003-3155-0817 Khubchandanı Jyotı Vishwakarma Institute of Technology https://orcid.org/0000-0003-2002-8775 Khubchandanı Payal Vishwakarma Institute of Technology 04 07 2026 16 4 472 481 03 14 2025 07 23 2025 Copyright © 2010, TWMS Journal of Applied and Engineering Mathematics 2010 TWMS Journal of Applied and Engineering Mathematics

In this paper we define fuzzy weak essential submodules to introduce the concept of fuzzy $\mathcal{W}$-closed submodules of an R-module $M$. Further, we define fuzzy fully semiprime module. We use the condition of fuzzy fully semiprime module to show that a non-constant fuzzy closed submodule is $\mathcal{W}$-closed in $M$. Also, the chain condition on fuzzy $\mathcal{W}$-closed submodules is studied.

 fuzzy $\mathcal{W}$-closed submodules fuzzy weak essential submodules fuzzy fully semi-prime submodules [1] Abbas, H. H. and Al-Aeashi, S. N., (2012), A fuzzy semi-essential submodules of a fuzzy module, J. of Kufa for Math. and Computer, 1 (5), pp. 31-37. [2] Acar, U., (2025), On L-fuzzy prime submodules, Hacettepe Journal of Mathematics and Statistics, 34, pp. 17-25. [3] Atani, S. E. and Sarai, F. E. K., (2017), On L-fuzzy multiplication modules, Discuss. Math. Gen. Algebra Appl., 37, pp. 209-221. [4] Hadi, I. M. A., (2004), Semiprime fuzzy submodules of fuzzy modules, Al-Haitham J. of Pure and Appl. Sci., 17 (3), pp. 112-123. [5] Hassan, K. M. and Hatam Y. Khalaf., (2020), Weak essential fuzzy submodules of fuzzy modules, Ibn Al-Haitham Jour. for Pure and Appl. Sci., 33 (4), pp. 65-72. [6] Hassan, K. M. and Hatam Y. Khalaf., (2020), Some properties of the essential fuzzy and closed fuzzy submodules, Iraqi Journal of Science, 61 (4), pp. 890-897. [7] Khubchandani, P. and Khubchandani, J., (2024), Fuzzy bioatomic pairs in fuzzy lattices, TWMS J. App. and Eng. Math., 14 (4), pp. 1485-1494. [8] Khubchandani, P. and Khubchandani, J., (2025), Characterization of fuzzy ⊥-⊤ distriutive lattice in fuzzy lattices, TWMS J. App. and Eng. Math., 15 (2), pp. 377-386. [9] Khubchandani, P. and Khubchandani, J., (2025), Fuzzy perspectivity in fuzzy lattices, Journal of Algebraic Systems, 13 (3), pp. 175-184. [10] Mohammed, M. Ali Radman Al-Shamiri., (2015), On fuzzy semiprime submodules, International Journal of Innovation and Applied Studies, 13 (4), pp. 929-934. [11] Moderson, J. N. and Malik. D. S., (1998), Fuzzy commutative algebra, World scientific, River Edge, NJ, USA. [12] Nimbhorkar, S. K. and Khubchandani, J. A., (2020), Fuzzy essential-small submodules and fuzzy small-essential submodules, Journal of hyperstructures, 9 (2), pp. 52-67. [13] Ahmmad J., Mahmood T., Rehman U.Ur, (2025) Market Segmentation Strategies Based on Complex Fuzzy Rough Dombi Aggregation Operators, Appl. Comput. Math., Vol.24, No.1, 2025, pp.38-61. [14] Nimbhorkar, S. K. and Khubchandani, J. A., (2023), Fuzzy semi-essential submodules and fuzzy semi-closed submodules, TWMS J. App and Eng. Math., 13 (2), pp. 568-575. [15] Nimbhorkar, S. K. and Khubchandani J. A., (2024), L-Fuzzy hollow modules and L-fuzzy multiplication modules, Kragujevac Journal of Mathematics, 48 (3), pp. 423-432. [16] Saikia, H. K. and Kalita, M. C., (2007), On fuzzy essential submodules, The Journal of Fuzzy Mathematics, 17 (1), pp. 109-117. [17] Talee, B., (2019), Intuitionistic fuzzy small submodules and their properties, Fuzzy Information and Engineering, 11 (3), pp. 307-319. [18] Talee, B., Oskooie, B. and Davaz, B., (2024), Some properties of intuitionistic fuzzy modules, J. Fuzzy Ext. Appl., 5 (2), pp. 190-198. [19] Wu X., Zhu Z., Wang T, Zhang X., (2025), Picture Fuzzy Interactional Aggregation Operators via Strict Triangular Norms and Applications to Multi-Criteria Decision Making, Appl. Comput. Math., Vol.24, No.1, 2025, pp.62-100 [20] Zadeh, L. A., (1965), Fuzzy sets, Inform. and Control, 8, pp. 338-353.