Algebraic properties of complex intuitionistic fuzzy submodules over $ \mathbb{R} $-modules

Sanaa Ahmed Bajri; Sarka Hoskova-Mayerova; Muhammad Haris Mateen; Kholood Alnefaie; Bijan Davvaz; Sanaa Ahmed Bajri; Sarka Hoskova-Mayerova; Muhammad Haris Mateen; Kholood Alnefaie; Bijan Davvaz

doi:10.3934/math.20261126

AIMS Mathematics

2026, Volume 11, Issue 9: 28273-28296. doi: 10.3934/math.20261126

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Research article

Algebraic properties of complex intuitionistic fuzzy submodules over $ \mathbb{R} $-modules

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics and Physics, University of Defence, Brno, 66210, Czech Republic
3.
School of Mathematics, Minhaj University Lahore, Lahore 54770, Pakistan
4.
Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia
5.
Department of Mathematical Sciences, Yazd University, Yazd, Iran

Received: 06 April 2026 Revised: 20 August 2026 Accepted: 24 August 2026 Published: 04 September 2026
MSC : 13E15, 08A72, 03E72I

Complex intuitionistic fuzzy sets ($ \mathbb{CIFS} $s) extend intuitionistic fuzzy sets ($ \mathbb{IFS} $s) by allowing membership and non-membership grades to take complex values, thus providing a more flexible framework for handling uncertainty. Motivated by the importance of module theory in modern algebra, this paper introduces the concept of complex intuitionistic fuzzy submodules of $ \mathbb{R} $-modules ($ \mathbb{CIFSM} $s$ \mathbb{R}) $ and investigates their structural properties. Fundamental notions including complex intuitionistic fuzzy quotient modules ($ \mathbb{CIFQM} $), level submodules, and supports are defined and characterized. Moreover, the behavior of complex intuitionistic fuzzy submodules under module homomorphisms ($ \mathbb{CIFSMH} $) is studied, and complex intuitionistic fuzzy versions of the fundamental homomorphism theorems for modules are established. The results extend existing work on intuitionistic fuzzy algebraic structures and provide a foundation for further study of complex fuzzy algebraic systems.
Citation: Sanaa Ahmed Bajri, Sarka Hoskova-Mayerova, Muhammad Haris Mateen, Kholood Alnefaie, Bijan Davvaz. Algebraic properties of complex intuitionistic fuzzy submodules over $ \mathbb{R} $-modules[J]. AIMS Mathematics, 2026, 11(9): 28273-28296. doi: 10.3934/math.20261126

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Abstract

Complex intuitionistic fuzzy sets ($ \mathbb{CIFS} $s) extend intuitionistic fuzzy sets ($ \mathbb{IFS} $s) by allowing membership and non-membership grades to take complex values, thus providing a more flexible framework for handling uncertainty. Motivated by the importance of module theory in modern algebra, this paper introduces the concept of complex intuitionistic fuzzy submodules of $ \mathbb{R} $-modules ($ \mathbb{CIFSM} $s$ \mathbb{R}) $ and investigates their structural properties. Fundamental notions including complex intuitionistic fuzzy quotient modules ($ \mathbb{CIFQM} $), level submodules, and supports are defined and characterized. Moreover, the behavior of complex intuitionistic fuzzy submodules under module homomorphisms ($ \mathbb{CIFSMH} $) is studied, and complex intuitionistic fuzzy versions of the fundamental homomorphism theorems for modules are established. The results extend existing work on intuitionistic fuzzy algebraic structures and provide a foundation for further study of complex fuzzy algebraic systems.

References

[1]	E. Noether, Idealtheorie in Ringbereichen, Math. Ann., 83 (1921), 24–66. https://doi.org/10.1007/BF01464225 doi: 10.1007/BF01464225
[2]	H. Cartan, S. Eilenberg, Homological algebra, Princeton University Press, 1956. https://doi.org/10.1515/9781400883844
[3]	T. Y. Lam, Lectures on modules and rings, Springer, 189 (1999). https://doi.org/10.1007/978-1-4612-0525-8
[4]	J. Xing, L. Chen, M. Bossert, Progressive algebraic soft-decision decoding of Reed–Solomon codes using module minimization, IEEE T. Commun., 67 (2019), 7379–7391. https://doi.org/10.1109/TCOMM.2019.2927207 doi: 10.1109/TCOMM.2019.2927207
[5]	A. K. Sinha, Application of module structure of algebra in coding theory in different branches of engineering, Int. J. Sci. Eng. Tech., 2 (2013), 295–297.
[6]	L. A. Zadeh, Fuzzy sets information and control, Inform. Control, 8 (1965), 338–353. https://doi.org/10.2307/2272014 doi: 10.2307/2272014
[7]	A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512–517. https://doi.org/10.1016/0022-247X(71)90199-5 doi: 10.1016/0022-247X(71)90199-5
[8]	W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Set. Syst., 8 (1982), 133–139. https://doi.org/10.1016/0165-0114(82)90003-3 doi: 10.1016/0165-0114(82)90003-3
[9]	W. J. Liu, Operations on fuzzy ideals, Fuzzy Set. Syst., 11 (1983), 31–41. https://doi/abs/10.5555/2846983.2847100 doi: 10.5555/2846983.2847100
[10]	V. N. Niya, H. Babaei, A. Rezaei, On fuzzy sub-semi-rings of nexuses, AIMS Math., 9 (2024), 36140–36157. https://doi.org/10.3934/math.20241715 doi: 10.3934/math.20241715
[11]	C. V. Negoita, D. A. Ralescu, Applications of fuzzy sets to systems analysis, Birkhäuser, 1975. https://doi.org/10.1007/978-3-0348-5921-9 doi: 10.1007/978-3-0348-5921-9
[12]	S. R. Lopez-Permouth, D. S. Malik, On categories of fuzzy modules, Inform. Sci., 52 (1990), 211–220. https://doi.org/10.1016/0020-0255(90)90043-A doi: 10.1016/0020-0255(90)90043-A
[13]	M. Gulistan, S. Abdullah, S. Hussian, On anti fuzzy relations in modules, NTMSCI, 3 (2015), 58–64.
[14]	Y. B. Jun, K. Hur, K. J. Lee, Hyperfuzzy subalgebras of $BCK/BCI$-algebras, Ann. Fuzzy Math. Inform., 15 (2018), 17–28. https://doi.org/10.30948/afmi.2018.15.1.17 doi: 10.30948/afmi.2018.15.1.17
[15]	S. Z. Song, S. J. Kim, Y. B. Jun, Hyperfuzzy ideals in $BCK/BCI$-algebras, Mathematics, 5 (2017), 81. https://doi.org/10.3390/math5040081 doi: 10.3390/math5040081
[16]	H. Bordbar, M. R. Bordbar, R. A. Borzooei, Y. B. Jun, $N$-subalgebras of $BCK/BCI$-algebras which are induced from hyperfuzzy structures, Iran. J. Math. Sci. Inform., 16 (2021), 179–195. https://doi.org/10.29252/ijmsi.16.2.163 doi: 10.29252/ijmsi.16.2.163
[17]	K. Sun, X. Yuan, H. Li, Fuzzy hypergroups based on fuzzy relations, Comput. Math. Appl., 60 (2010), 610–622. https://doi.org/10.1016/j.camwa.2010.05.007 doi: 10.1016/j.camwa.2010.05.007
[18]	F. Smarandache, The SuperHyperFunction and the neutrosophic SuperHyperFunction, Neutrosophic Sets Syst., 49 (2022), 594–600. https://doi.org/10.5281/zenodo.6466524 doi: 10.5281/zenodo.6466524
[19]	H. Sankari, M. Abobala, $n$-refined neutrosophic modules, Neutrosophic Sets Syst., 36 (2020), 1–11. https://doi.org/10.5281/zenodo.4065342 doi: 10.5281/zenodo.4065342
[20]	P. K. Sharma, Anti fuzzy submodule of a module, Adv. Fuzzy Set. Syst., 12 (2012), 49–57.
[21]	H. Abbass, S. Al-Aeashi, A fuzzy module, J. Kufa Math. Comput., 1 (2012).
[22]	B. Davvaz, N. Firouzkouhi, Fundamental relation on fuzzy hypermodules, Soft Comput., 23 (2019), 13025–13033. https://doi.org/10.1007/s00500-019-04299-3 doi: 10.1007/s00500-019-04299-3
[23]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 1–137. https://doi.org/10.1007/978-3-7908-1870-3 doi: 10.1007/978-3-7908-1870-3
[24]	J. Jin, X. Yang, H. Du, D. Pamucar, A novel interval-valued intuitionistic fuzzy distance measure incorporating min/max interaction terms, J. Contemp. Decis. Sci., 2 (2026), 84–96. https://doi.org/10.67334/cds21202610 doi: 10.67334/cds21202610
[25]	A. Ishtiaq, K. T. Kubra, A. Uzair, A. Ali, Quantifying multi-cause psychological disorder risk through an advanced mathematical model using intuitionistic pentagonal fuzzy logic, Spectr. Oper. Res., 2025, 1–21. https://doi.org/10.31181/sor41202762 doi: 10.31181/sor41202762
[26]	T. Basuri, K. H. Gazi, S. G. Das, S. P. Mondal, Ranking higher education institutions using Entropy–VIKOR with generalized pentagonal intuitionistic fuzzy numbers, J. Contemp. Decis. Sci., 2 (2026), 64–83. https://doi.org/10.67334/cds2120266 doi: 10.67334/cds2120266
[27]	R. Biswas, Intuitionistic fuzzy subgroups, Math. Forum, 10 (1989), 86–94.
[28]	B. Davvaz, W. A. Dudek, Y. B. Jun, Intuitionistic fuzzy Hv-submodules, Inform. Sci., 176 (2006), 285–300. https://doi.org/10.1016/j.ins.2004.10.009 doi: 10.1016/j.ins.2004.10.009
[29]	M. Muslikh, O. D. Wijaya, A. R. Alghofari, N. Hidayat, The properties of intuitionistic anti fuzzy module t-norm and t conorm, CAUCHY-J. Mat. Murni dan Aplikasi, 7 (2022).
[30]	S. Bhunia, G. Ghorai, Q. Xin, On the characterization of Pythagorean fuzzy subgroups, AIMS Math., 6 (2021), 962–978. https://doi.org/10.3934/math.2021058 doi: 10.3934/math.2021058
[31]	A. Razaq, G. Alhamzi, On Pythagorean fuzzy ideals of a classical ring, AIMS Math., 8 (2023), 4280–4303. https://doi.org/10.3934/math.2023213 doi: 10.3934/math.2023213
[32]	P. K. Sharma, Direct sum of intuitionistic fuzzy submodules of G-module, Asian J. Fuzzy Appl. Math., 4 (2016).
[33]	N. Firouzkouhi, A. Amini, M. Nazari, F. Alkhatib, H. Bordbar, C. Cheng, et al., Advanced artificial intelligence system by intuitionistic fuzzy $\Gamma$-subring for automotive robotic manufacturing, Artif. Intell. Rev., 56 (2023), 9639–9664. https://doi.org/10.1007/s10462-023-10396-5 doi: 10.1007/s10462-023-10396-5
[34]	N. Firouzkouhi, A. Amini, C. Cheng, A. Zarrabi, B. Davvaz, Intuitionistic fuzzy set of $\Gamma$-submodules and its application in modeling spread of viral diseases, mutated COVID-$n$, via flights, Int. J. Intell. Syst., 37 (2022), 5134–5151. https://doi.org/10.1002/int.22754 doi: 10.1002/int.22754
[35]	N. Firouzkouhi, A. Amini, F. Alkhatib, A. Bani-Mustafa, C. Cheng, B. Davvaz, Global e-Commerce (GeC) theory by intuitionistic fuzzy $\Gamma$-submodule, Eng. Appl. Artif. Intell., 123 (2023). https://doi.org/10.1016/j.engappai.2023.106252 doi: 10.1016/j.engappai.2023.106252
[36]	D. Ramot, R. Milo, M. Friedman, A. Kandel, Complex fuzzy sets, IEEE T. Fuzzy Syst., 10 (2002), 171–186. https://doi.org/10.1109/91.995119 doi: 10.1109/91.995119
[37]	D. Ramot, M. Friedman, G. Langholz, A. Kandel, Complex fuzzy logic, IEEE T. Fuzzy Syst., 11 (2003), 450–461. https://doi.org/10.1109/TFUZZ.2003.814832 doi: 10.1109/TFUZZ.2003.814832
[38]	A. Al-Husban, A. R. Salleh, Complex fuzzy group based on complex fuzzy space, Global J. Pure Appl. Math., 12 (2016), 1433–1450.
[39]	A. A. Alharbi, D. Alghazzawi, Some characterizations of certain complex fuzzy subgroups, Symmetry, 14 (2022), 1812. https://doi.org/10.3390/sym14091812 doi: 10.3390/sym14091812
[40]	H. Imtiaz, U. Shuaib, A. Razaq, J. B. Liu, Applications of conjunctive complex fuzzy subgroups to Sylow theory, AIMS Math., 9 (2024), 38–54. https://doi.org/10.3934/math.2024003 doi: 10.3934/math.2024003
[41]	A. Imtiaz, U. Shuaib, H. Alolaiyan, A. Razaq, M. Gulistan, On structural properties of $\lambda$-complex fuzzy sets and their applications, Complexity, 2020 (2020), 2038724. https://doi.org/10.1155/2020/2038724 doi: 10.1155/2020/2038724
[42]	A. Ali, M. H. Mateen, Q. Xin, T. Alsuraiheed, G. Alhamzi, $(\epsilon, \delta)$-complex anti fuzzy subgroups and their applications, AIMS Math., 9 (2024), 11580–11595. https://doi.org/10.3934/math.2024568 doi: 10.3934/math.2024568
[43]	B. Almutairi, A. Ali, Q. Xin, A. Khan, A novel concept of complex anti-fuzzy isomorphism over groups, Symmetry, 15 (2023), 1693. https://doi.org/10.3390/sym15091693 doi: 10.3390/sym15091693
[44]	A. Alkouri, A. R. Salleh, Complex Atanassov's intuitionistic fuzzy sets, AIP Conf. Proc., 1482 (2012), 287382. https://doi.org/10.1155/2013/287382 doi: 10.1155/2013/287382
[45]	A. Razzaque, A. Razaq, A. Khalid, I. Masmali, U. Shuaib, G. Alhamzi, Selecting an optimal approach to reduce drivers of climate change in a complex intuitionistic fuzzy environment, Sustainability, 15 (2023), 12300. https://doi.org/10.3390/su151612300 doi: 10.3390/su151612300
[46]	K. Ullah, N. Rehman, A. Ali, A robust circular complex intuitionistic fuzzy framework for optimizing agricultural robot decision-making under uncertain environmental conditions, Spectr. Decis. Mak. Appl., 4 (2026), 1–42. https://doi.org/10.31181/sdmap41202770 doi: 10.31181/sdmap41202770
[47]	V. Shahin, M. Alimohammadlou, D. Pamucar, An interval-valued circular intuitionistic fuzzy MARCOS method for renewable energy source selection, Spectr. Decis. Mak. Appl., 3 (2026), 243–268. https://doi.org/10.31181/sdmap31202645 doi: 10.31181/sdmap31202645
[48]	M. Gulzar, M. H. Mateen, D. Alghazzawi, N. Kausar, A novel applications of complex intuitionistic fuzzy sets in group theory, IEEE Access, 8 (2020). https://doi.org/10.1109/ACCESS.2020.3034626 doi: 10.1109/ACCESS.2020.3034626
[49]	Z. Wu, Z. Gong, Algebraic structure of some complex intuitionistic fuzzy subgroups and their homomorphism, AIMS Math., 10 (2025), 4067–4091. https://doi.org/10.3934/math.2025189 doi: 10.3934/math.2025189
[50]	D. Al-Sharoa, $(\alpha_{1}, 2, \beta_{1}, 2)$-complex intuitionistic fuzzy subgroups and its algebraic structure, AIMS Math., 8 (2023), 8082–8116. https://doi.org/10.3934/math.2023409 doi: 10.3934/math.2023409
[51]	H. Alolaiyan, H. A. Alshehri, M. H. Mateen, D. Pamucar, M. Gulzar, A novel algebraic structure of $(\alpha, \beta)$-complex fuzzy subgroups, Entropy, 23 (2021), 992. https://doi.org/10.3390/e23080992 doi: 10.3390/e23080992
[52]	M. Jawad, N. Nigar, S. Hoskova-Mayerova, B. Davvaz, M. H. Mateen, Fundamental theorems of group isomorphism under the framework of complex intuitionistic fuzzy set, AIMS Math., 10 (2025), 1900–1920. https://doi.org/10.3934/math.2025088 doi: 10.3934/math.2025088
[53]	M. Jawad, S. Hoskova-Mayerova, N. Nigar, M. H. Mateen, An innovative method for employing complex intuitionistic fuzzy ideals in algebras, Eur. J. Pure Appl. Math., 18 (2025), 6798–6798. https://doi.org/10.29020/nybg.ejpam.v18i4.6798 doi: 10.29020/nybg.ejpam.v18i4.6798
[54]	M. Jawad, S. Hoskova-Mayerova, N. Nigar, S. A. Bajri, M. H. Mateen, C. Jana, Incorporating £-complex intuitionistic fuzzy set to Sylow theorems in group theory, J. Math., 2026, 6650937. https://doi.org/10.1155/jom/6650937 doi: 10.1155/jom/6650937
[55]	M. Gulzar, M. H. Mateen, Y. M. Chu, D. Alghazzawi, G. Abbas, Generalized direct product of complex intuitionistic fuzzy subrings, Int. J. Comput. Intell. Syst. 14 (2021), 582–593. https://doi.org/10.2991/ijcis.d.210106.001 doi: 10.2991/ijcis.d.210106.001
[56]	K. Alnefaie, S. Hoskova-Mayerova, M. H. Mateen, B. Davvaz, Incorporating complex bipolar fuzzy set with subrings and application in decision making, AIMS Math., 10 (2025), 27073–27102. https://doi.org/10.3934/math.20251190 doi: 10.3934/math.20251190
[57]	F. Smarandache, A unifying field in logics: Neutrosophic logic, neutrosophy, neutrosophic set, and neutrosophic probability, Infinite Study, Phoenix, AZ, USA, 2005.
[58]	A. Y. Hummdi, A. Elrawy, On neutrosophic ideals and prime ideals in rings, AIMS Math., 9 (2024), 24762–24775. https://doi.org/10.3934/math.20241205 doi: 10.3934/math.20241205
[59]	V. Çetkin, B. P. Varol, H. Aygün, On neutrosophic submodules of a module, Hacet. J. Math. Stat., 46 (2017), 791–799. https://doi.org/10.15672/HJMS.2017437 doi: 10.15672/HJMS.2017437
[60]	G. Muhiuddin, N. Abughazalah, B. Elavarasan, K. Porselvi, D. Al-Kadi, Neutrosophic $\mathcal{N}$-structures in semimodules over semirings, Symmetry, 16 (2024), 41. https://doi.org/10.3390/sym16010041 doi: 10.3390/sym16010041
[61]	M. Ali, F. Smarandache, Complex neutrosophic set, Neural Comput. Appl., 28 (2017), 1817–1834. https://doi.org/10.1007/s00521-015-2154-y doi: 10.1007/s00521-015-2154-y
[62]	M. Gulistan, A. Khan, A. Abdullah, N. Yaqoob, Complex neutrosophic subsemigroups and ideals, Int. J. Anal. Appl., 16 (2018), 97–116. https://doi.org/10.28924/2291-8639-16-2018-97 doi: 10.28924/2291-8639-16-2018-97
[63]	T. Alsuraiheed, U. ur Rehman, M. A. Khan, T. Mahmood, Bipolar complex fuzzy submodules, Phys. Scripta, 99 (2024), 065225. https://doi.org/10.1088/1402-4896/ad421a/meta doi: 10.1088/1402-4896/ad421a/meta
[64]	Z. Khan, F. Hussain, I. Ullah, T. Rahim, M. Khan, R. Jan, et al., Some novel properties of complex intuitionistic fuzzy ideals in classical ring, Part. Differ. Equ. Appl. Math., 15 (2025). https://doi.org/10.1016/j.padiff.2024.100811 doi: 10.1016/j.padiff.2024.100811
[65]	A. Khamis, A. G. Ahmad, A note on direct product of complex intuitionistic fuzzy subfield, J. Intell. Fuzzy Syst., 45 (2023), 2111–2132. https://doi.org/10.3233/JIFS-230597 doi: 10.3233/JIFS-230597
[66]	M. H. Mateen, T. Alsuraiheed, H. Mohamed, A new algebraic structure of complex Pythagorean fuzzy subfield, IEEE Access, 2023, 145636–145650. https://doi.org/10.1109/ACCESS.2023.3346188 doi: 10.1109/ACCESS.2023.3346188
[67]	M. O. Alsarahead, A. G. Ahmad, Complex intuitionistic fuzzy subrings, Borneo Sci., 38 (2017), 24–37.
[68]	P. K. Singh, Complex plithogenic set, Int. J. Neutrosophic Sci., 18 (2022), 57–72. https://doi.org/10.54216/IJNS.180106 doi: 10.54216/IJNS.180106

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