On 1-Absorbing Fuzzy Ideals of Commutative Semirings

Abstract

In this study, the algebraic structure of 1-absorbing ideals is first examined and applied to fuzzy sets, along with an investigation into the relationships and algebraic properties between them. The contribution to this work's literature involves examining 1-absorbing fuzzy primary ideals. Features of 1-absorbing fuzzy primary ideals are explored, and it is demonstrated, for instance, that I is deemed a 1-absorbing fuzzy primary ideal of P if I is a fuzzy primary ideal of P. Additionally, I is considered a 2-absorbing fuzzy primary ideal of P if I is a 1-absorbing fuzzy primary ideal of P. Furthermore, these theorems are elucidated through specific examples.

Keywords

• Fuzzy sets
• Fuzzy ideals
• 1-absorbing ideals
• 1-absorbing fuzzy ideals
• 1-absorbing fuzzy primary ideals

References

1. [1] Atani, R. E., and Atani, S. E., “Ideal theory in commutative semirings”, Buletinul Academiei de Stiinte a Republicii Moldova Matematica 2 (2008) : 14-23.
2. [2] Zadeh, L.A., “Fuzzy sets”, Inform. and Control 8 (1965) : 338-353.
3. [3] Roberts, D. W., “Ordination based on fuzzy set theory”, Vegetation 66(3) (1986) : 123-131.
4. [4] Zimmermann, H. J., “Fuzzy set theory”, Wiley interdisciplinary reviews: Computational statistics 2(3) (2010) : 317-332.
5. [5] Chang, C. L., “Fuzzy topological spaces”, Journal of Mathematical Analysis and Applications 24(1) (1968) : 182-190.
6. [6] Rosenfeld, A., “Fuzzy groups”, Journal of Mathematical Analysis and applications 35(3) (1971) : 512-517.
7. [7] Liu, W. J., “Fuzzy invariant subgroups and fuzzy ideals”, Fuzzy sets and Systems 8(2) (1982) : 133-139.
8. [8] Abou-Zaid, S., “On fuzzy subnear-rings and ideals”, Fuzzy sets and systems 44(1) (1991) : 139-146.

9. [9] Atanassov, K. T., “Intuitionistic fuzzy sets. In Intuitionistic fuzzy sets”, Physica, Heidelberg (1999) : 1-137.
10. [10] Szmidt, E., and Kacprzyk, J., “Distances between intuitionistic fuzzy sets”, Fuzzy sets and systems 114(3) (2000) : 505-518.
11. [11] Özdemir, O., & Kalınkara, Y., “Bulanık Mantık: 2000-2020 Yılları Arası Tez ve Makale Çalışmalarına Yönelik Bir İçerik Analizi”, Acta Infologica 4(2) (2020) : 155-174.
12. [12] Badawi, A., “On 2-absorbing ideals of commutative rings”, Bulletin of the Australian Mathematical Society 75(3) (2007) : 417-429.
13. [13] Badawi, A., & Celikel, E. Y., “On 1-absorbing primary ideals of commutative rings”, Journal of Algebra and Its Applications 19(06) (2020) : 2050111.
14. [14] Yassine, A., Nikmehr, M. J., & Nikandish, R., “On 1-absorbing prime ideals of commutative rings”, Journal of Algebra and its Applications 20(10) (2021): 2150175.
15. [15] Mandal, D., “On 2-absorbing fuzzy ideals of commutative semirings”, TWMS Journal of Applied and Engineering Mathematics 11(2) (2021) : 368.
16. [16] Darani, A.Y., Hashempoor, A., “L-fuzzy 0-(1-or 2-or 3-) 2-absorbing ideals in semiring”, Ann. Fuzzy Math. Inform. 7 (2014) : 303-311.
17. [17] Jonathan S. Golan, “Semirings and their applications”, Springer (2013) : 105-120.
18. [18] Darani, Ahmad Yousefian, “On 2-absorbing and weakly 2-absorbing ideals of commutative semirings”, Kyungpook Mathematical Journal 52 (1) (2012) : 91-97.
19. [19] Chaudhari, Jayprakash Ninu, “2-absorbing ideals in semirings”, International Journal of Al- gebra 6 (6) (2012) : 265-270.
20. [20] Groenewald, Nico, “On weakly 2-absorbing ideals of non-commutative rings”, Afrika Matematika 32(7-8) (2021) : 1669-1683.
21. [21] Soheilnia, Fatemeh, “On 2-absorbing and weakly 2-absorbing primary ideals of a commutative semiring”, Kyungpook Mathematical Journal 56 (1) (2016).
22. [22] Behzadipour, Hussein, and Peyman Nasehpour, “On 2-absorbing ideals of commutative semir- ings”, Journal of Algebra and Its Applications 19 (2)(2020) : 2050034.
23. [23] Jaber, Ameer., “Properties of weakly 2-absorbing primal ideals”, Italian Journal of Pure and Applied Mathematics 7 (2022) : 609-619.
24. [24] Groenewald, N. J., “On 2-absorbing and weakly 2-absorbing principally right primary ideals”, Journal of Algebra and Related Topics 9 (2) (2021) : 47-67.
25. [25] Sahoo, Taptee, Deepak Shetty, M., Groenewald, N. J., Harikrishnan, P. K., Kuncham, S. P., “On completely 2-absorbing ideals of N-groups”, Journal of Discrete Mathematical Sciences and Cryptography 24 (2) (2021) : 541-556.
26. [26] Yassine, A., M. J. Nikmehr, and R. Nikandish., “On 1-absorbing prime ideals of commutative rings”, Journal of Algebra and its Applications 20(10) (2021) : 2150175.
27. [27] Koc, Suat, Ünsal Tekir, and Eda Yıldız, “On weakly 1-absorbing prime ideals”, Ricerche di Matematica 72(2) (2023) : 723-738.
28. [28] Groenewald, Nico., “1-absorbing prime ideals and weakly 1-absorbing prime ideals in non-commutative rings”, S˜ao Paulo Journal of Mathematical Sciences (2022) : 1-17.
29. [29] Bouba, E. M., Tamekkante, M., Tekir, U¨ ., and Ko¸c, S., “Notes on 1-absorbing prime ideals”,Proceedings of the Bulgarian Academy of Sciences 75(5) (2022) : 631-639.
30. [30] Abu-Dawwas, R., Yıldız, E., Tekir, U¨ ., Koc, S., “On graded 1-absorbing prime ideals.” Sao Paulo Journal of Mathematical Sciences 15 (2021) : 450-462.
31. [31] Anbarloei, Mahdi., “On 1-absorbing prime hyperideal and some of its generalizations”, Journal of Mathematics (2022) : 4947019.
32. [32] Saleh, Mohammad, and Ibaa M., “On weakly 1-absorbing primary ideals of commutative semirings”, Communications in Advanced Mathematical Sciences 5(4) (2022) : 199-208.