This article aims to introduce and explore the concept of neutrosophic ideals in the context of rings. Besides, we investigate how the property of regularity in a ring can be understood through the lens of neutrosophic ideals. We further present the concept of neutrosophic prime ideals and systematically identify all neutrosophic prime ideals in $ \mathbb{Z} $.
Citation: Ali Yahya Hummdi, Amr Elrawy. On neutrosophic ideals and prime ideals in rings[J]. AIMS Mathematics, 2024, 9(9): 24762-24775. doi: 10.3934/math.20241205
This article aims to introduce and explore the concept of neutrosophic ideals in the context of rings. Besides, we investigate how the property of regularity in a ring can be understood through the lens of neutrosophic ideals. We further present the concept of neutrosophic prime ideals and systematically identify all neutrosophic prime ideals in $ \mathbb{Z} $.
[1] |
N. A. Alreshidi, Z. Shah, M. J. Khan, Similarity and entropy measures for circular intuitionistic fuzzy sets, Eng. Appl. Artif. Intel., 131 (2024), 107786. https://doi.org/10.1016/j.engappai.2023.107786 doi: 10.1016/j.engappai.2023.107786
![]() |
[2] |
K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
![]() |
[3] | D. M. Burro, A first course in ring and ideals, New York: Addison-Wesley Educational Publishers Inc, 1970. |
[4] |
V. Cetkin, H. Aygün, An approach to neutrosophic subgroup and its fundamental properties, J. Intell. Fuzzy Syst., 29 (2015), 1941–1947. https://doi.org/10.3233/IFS-151672 doi: 10.3233/IFS-151672
![]() |
[5] |
V. Cetkin, H. Aygün, An approach to neutrosophic ideals, Universal Journal of Mathematics and Applications, 1 (2018), 132–136. https://doi.org/10.32323/ujma.383115 doi: 10.32323/ujma.383115
![]() |
[6] | V. Cetkin, H. Aygün, An approach to neutrosophic subrings, Sakarya University Journal of Science, 23 (2019), 472–477. |
[7] |
A. Elrawy, M. Abdalla, Results on a neutrosophic sub-rings, AIMS Mathematics, 8 (2023), 21393–21405. https://doi.org/10.3934/math.20231090 doi: 10.3934/math.20231090
![]() |
[8] |
A. Elrawy, M. Abdalla, M. A. Saleem, Some results on neutrosophic group, J. Math., 2023 (2023), 4540367. https://doi.org/10.1155/2023/4540367 doi: 10.1155/2023/4540367
![]() |
[9] |
H. Garg, A new exponential-logarithm-based single-valued neutrosophic set and their applications, Expert Syst. Appl., 238 (2024), 121854. https://doi.org/10.1016/j.eswa.2023.121854 doi: 10.1016/j.eswa.2023.121854
![]() |
[10] |
Y. H. Guo, A. I. Shahin, H. Garg, An indeterminacy fusion of encoder-decoder network based on neutrosophic set for white blood cells segmentation, Expert Syst. Appl., 246 (2024), 123156. https://doi.org/10.1016/j.eswa.2024.123156 doi: 10.1016/j.eswa.2024.123156
![]() |
[11] |
A. Y. Hummdi, M. Abdalla, A. Elrawy, Neutrosophic sub-group over t-norm and t-co-norm, Axioms, 13 (2024), 101. https://doi.org/10.3390/axioms13020101 doi: 10.3390/axioms13020101
![]() |
[12] |
Y. B. Jun, Neutrosophic subalgebras of several types in BCK/BCI-algebras, Annals of Fuzzy Mathematics and Informatics, 14 (2017), 75–86. https://doi.org/10.30948/afmi.2017.14.1.75 doi: 10.30948/afmi.2017.14.1.75
![]() |
[13] | W. B. V. Kandasamy, F. Smarandache, Basic neutrosophic algebraic structures and their application to fuzzy and neutrosophic models, New York: ACM Press, 2004. |
[14] | W. B. V. Kandasamy, F. Smarandache, Neutrosophic rings, Arizona: Hexis, 2006. |
[15] | G. Kemper, A course in commutative algebra, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-03545-6 |
[16] | M. A. Ozturk, Y. B. Jun, Neutrosophic ideals in BCK/BCI-algebras based on neutrosophic points, Journal of the International Mathematical Virtual Institute, 8 (2018), 1–17. |
[17] |
Y. Saber, F. Alsharari, F. Smarandache, On single-valued neutrosophic ideals in $\check{S}$ostak sense, Symmetry, 12 (2020), 193. https://doi.org/10.3390/sym12020193 doi: 10.3390/sym12020193
![]() |
[18] | A. A. Salama, F. Smarandache, Neutrosophic ideal theory neutrosophic local function and generated neutrosophic topology, Neutrosophic Theory and Its Applications, 2014 (2014), 213–218. |
[19] | F. Smarandache, Neutrosophy: neutrosophic probability, set, and logic: analytic synthesis & synthetic analysis, Rehoboth: American Research Press, 1998. |
[20] | F. Smarandache, Neutrosophic set–A generalization of the intuitionistic fuzzy set, International Journal of Pure and Applied Mathematics, 24 (2005), 287. |
[21] | F. Smarandache, Neutrosophic perspectives: triplets, duplets, multisets, hybrid operators, modal logic, hedge algebras, and applications, Brussels: Pons Publishing House, 2017. |
[22] |
J. V. Neumann, On regular rings, Proceedings of the National Academy of Sciences of the United States of America, 22 (1936), 707–713. https://doi.org/10.1073/pnas.22.12.707 doi: 10.1073/pnas.22.12.707
![]() |
[23] |
J. Q. Wang, X. H. Zhang, Intuitionistic fuzzy granular matrix: novel calculation approaches for intuitionistic fuzzy covering-based rough sets, Axioms, 13 (2024), 411. https://doi.org/10.3390/axioms13060411 doi: 10.3390/axioms13060411
![]() |
[24] |
L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
![]() |