The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set and is an effective approach of handling uncertain situations. Ring theory is a prominent branch of abstract algebra, vibrant in wide areas of current research in mathematics, computer science and mathematical/theoretical physics. In the theory of rings, the study of ideals is significant in many ways. Keeping in mind the importance of ring theory and Pythagorean fuzzy set, in the present article, we characterize the concept of Pythagorean fuzzy ideals in classical rings and study its numerous algebraic properties. We define the concept of Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal and prove that the set of all Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal forms a ring under certain binary operations. Furthermore, we present Pythagorean fuzzy version of the fundamental theorem of ring homomorphism. We also introduce the concept of Pythagorean fuzzy semi-prime ideals and give a detailed exposition of its different algebraic characteristics. In the end, we characterized regular rings by virtue of Pythagorean fuzzy ideals.
Citation: Abdul Razaq, Ghaliah Alhamzi. On Pythagorean fuzzy ideals of a classical ring[J]. AIMS Mathematics, 2023, 8(2): 4280-4303. doi: 10.3934/math.2023213
The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set and is an effective approach of handling uncertain situations. Ring theory is a prominent branch of abstract algebra, vibrant in wide areas of current research in mathematics, computer science and mathematical/theoretical physics. In the theory of rings, the study of ideals is significant in many ways. Keeping in mind the importance of ring theory and Pythagorean fuzzy set, in the present article, we characterize the concept of Pythagorean fuzzy ideals in classical rings and study its numerous algebraic properties. We define the concept of Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal and prove that the set of all Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal forms a ring under certain binary operations. Furthermore, we present Pythagorean fuzzy version of the fundamental theorem of ring homomorphism. We also introduce the concept of Pythagorean fuzzy semi-prime ideals and give a detailed exposition of its different algebraic characteristics. In the end, we characterized regular rings by virtue of Pythagorean fuzzy ideals.
| [1] | L. A. Zadeh, Fuzzy sets, In: Fuzzy sets, fuzzy logic, and fuzzy systems, 1996,394–432. https://doi.org/10.1142/9789814261302_0021 |
| [2] |
K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
|
| [3] |
F. Feng, H. Fujita, M. I. Ali, R. R. Yager, X. Y. Liu, Another view on generalized intuitionistic fuzzy soft sets and related multiattribute decision making methods, IEEE Trans. Fuzzy Syst., 27 (2019), 474–488. https://doi.org/10.1109/TFUZZ.2018.2860967 doi: 10.1109/TFUZZ.2018.2860967
|
| [4] |
F. Feng, Z. S. Xu, H. Fujita, M. Q. Liang, Enhancing PROMETHEE method with intuitionistic fuzzy soft sets, Int. J. Intel. Syst., 35 (2020), 1071–1104. https://doi.org/10.1002/int.22235 doi: 10.1002/int.22235
|
| [5] | R. R. Yager, Pythagorean fuzzy subsets, In: 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS), 2013, 57–61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375 |
| [6] |
X. L. Zhang, Z. S. Xu, Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets, Int. J. Intel. Syst., 29 (2014), 1061–1078. https://doi.org/10.1002/int.21676 doi: 10.1002/int.21676
|
| [7] |
F. Y. Xiao, W. P. Ding, Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis, Appl. Soft Comput., 79 (2019), 254–267. https://doi.org/10.1016/j.asoc.2019.03.043 doi: 10.1016/j.asoc.2019.03.043
|
| [8] |
Z. Wang, F. Y. Xiao, Z. H. Cao, Uncertainty measurements for Pythagorean fuzzy set and their applications in multiple-criteria decision making, Soft Comput., 26 (2022), 9937–9952. https://doi.org/10.1007/s00500-022-07361-9 doi: 10.1007/s00500-022-07361-9
|
| [9] | N. Bourbaki, Elements of the history of mathematics, Berlin, Heidelberg: Springer, 1994. |
| [10] | Bustomi, A. P. Santika, D. Suprijanto, Linear codes over the ring ${{\rm{Z}}_4} + {\rm{u}}{{\rm{Z}}_4} + {\rm{v}}{{\rm{Z}}_4} + {\rm{w}}{{\rm{Z}}_4} + {\rm{uv}}{{\rm{Z}}_4} + {\rm{uw}}{{\rm{Z}}_4} + {\rm{vw}}{{\rm{Z}}_4} + {\rm{uvw}}{{\rm{Z}}_4}$, IAENG Int. J. Comput. Sci., 48 (2021), 686–696. |
| [11] | N. Deo, Graph theory with applications to engineering and computer science, Mineola, New York: Dover Publications, 2016. |
| [12] |
A. Razaq, Iqra, M. Ahmad, M. A. Yousaf, S. Masood, A novel finite rings based algebraic scheme of evolving secure S-boxes for images encryption, Multimed. Tools Appl., 80 (2021), 20191–20215. https://doi.org/10.1007/s11042-021-10587-8 doi: 10.1007/s11042-021-10587-8
|
| [13] |
W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets Syst., 8 (1982), 133–139. https://doi.org/10.1016/0165-0114(82)90003-3 doi: 10.1016/0165-0114(82)90003-3
|
| [14] |
W. J. Liu, Operations on fuzzy ideals, Fuzzy Sets Syst., 11 (1983), 31–39. https://doi.org/10.1016/S0165-0114(83)80067-0 doi: 10.1016/S0165-0114(83)80067-0
|
| [15] |
D. S. Malik, Fuzzy ideals of Artinian rings, Fuzzy Sets Syst., 37 (1990), 111–115. https://doi.org/10.1016/0165-0114(90)90069-I doi: 10.1016/0165-0114(90)90069-I
|
| [16] | K. Hur, H. W. Kang, H. K. Song, Intuitionistic fuzzy subgroups and subrings, Honam Math. J., 25 (2003), 19–41. |
| [17] |
A. Altassan, M. H. Mateen, D. Pamucar, On fundamental theorems of fuzzy isomorphism of fuzzy subrings over a certain algebraic product, Symmetry, 13 (2021), 998. https://doi.org/10.3390/sym13060998 doi: 10.3390/sym13060998
|
| [18] |
M. Shabir, A. N. Al-Kenani, F. Javed, S. Bashir, An efficient approach to approximate fuzzy ideals of semirings using bipolar techniques, Mathematics, 10 (2022), 1009. https://doi.org/10.3390/math10071009 doi: 10.3390/math10071009
|
| [19] |
S. Hoskova-Mayerova, M. Al Tahan, Anti-fuzzy multi-ideals of near ring, Mathematics, 9 (2021), 494. https://doi.org/10.3390/math9050494 doi: 10.3390/math9050494
|
| [20] |
M. Al Tahan, S. Hoskova-Mayerova, B. Davvaz, An approach to fuzzy multi-ideals of near rings, J. Intel. Fuzzy Syst., 41 (2021), 6233–6243. https://doi.org/10.3233/JIFS-202914 doi: 10.3233/JIFS-202914
|
| [21] |
X. L. Xin, Y. L. Fu, Fuzzy Zorn's lemma with applications, Appl. Math. J. Chinese Univ., 36 (2021), 521–536. https://doi.org/10.1007/s11766-021-4043-8 doi: 10.1007/s11766-021-4043-8
|
| [22] |
G. M. Addis, N. Kausar, M. Munir, Fuzzy homomorphism theorems on rings, J. Discrete Math. Sci. Cryptogr., 25 (2022), 1757–1776. https://doi.org/10.1080/09720529.2020.1809777 doi: 10.1080/09720529.2020.1809777
|
| [23] |
S. Bhunia, G. Ghorai, Q. Xin, M. Gulzar, On the algebraic attributes of (α, β)-Pythagorean fuzzy subrings and (α, β)-Pythagorean fuzzy ideals of rings, IEEE Access, 10 (2022), 11048–11056. https://doi.org/10.1109/ACCESS.2022.3145376 doi: 10.1109/ACCESS.2022.3145376
|
| [24] |
A. Hakim, H. Khan, I. Ahmad, A. Khan, Fuzzy bipolar soft semiprime ideals in ordered semigroups, Heliyon, 7 (2021), e06618. https://doi.org/10.1016/j.heliyon.2021.e06618 doi: 10.1016/j.heliyon.2021.e06618
|
| [25] |
N. Kausar, B. U. Islam, M. Y. Javaid, S. A. Ahmad, U. Ijaz, Characterizations of non-associative rings by the properties of their fuzzy ideals, J. Taibah Univ. Sci., 13 (2019), 820–833. https://doi.org/10.1080/16583655.2019.1644817 doi: 10.1080/16583655.2019.1644817
|
| [26] | D. M. Burton, A first course in ring and ideals, Addison-Wesley, 1970. |
Abdul Razaq, Ghaliah Alhamzi. On Pythagorean fuzzy ideals of a classical ring[J]. AIMS Mathematics, 2023, 8(2): 4280-4303. doi: 10.3934/math.2023213
DownLoad: