A New Approach to Soft Prime Int-Ideal: Soft Primary Int-Ideal and Soft 1-Absorbing Primary Int-Ideal
Abstract
In this paper, we introduce some new structures such as soft primary int-ideal and soft 1-absorbing primary int-ideal, which are different generalizations of the soft prime int-ideal. We study these new structures theoretically and practically. We examine the relationship between these new structures. One of our most important results is that a soft prime int-ideal is both soft primary int-ideal and soft $1$-absorbing primary int-ideal. It is shown with the help of examples that opposites are not always true. Moreover, We show that every soft primary int-ideal is a soft 1-absorbing primary int-ideal, but the opposite of this situation is not validated. Then, another result is the behavior of the soft radical on these structures. Besides, we strengthen this structure through various problems. Moreover, we determine that soft homomorphic images and pre-images of these new structures are guarded under onto ring homomorphisms.
Keywords
Soft set soft int-ideal soft prime int-ideal soft primary int-ideal soft 1-absorbing primary int-ideal
References
- L. A. Zadeh, Fuzzy sets, Information and Control 8 (3) (1965) 338-353.
- Z. Pawlak, Rough sets, Journal of Computer and Information Sciences 11 (5) (1982) 341-356.
- D. A. Molodstov, Soft set theory-first results, Computers and Mathematics with Applications 37 (4-5) (1999) 19-31.
- P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Computers and Mathematics with Applications 45 (4-5) (2003) 555-562.
- M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, On some new operations in soft set theory, Computers and Mathematics with Applications 57 (9) (2009) 1547-1553.
- M. I. Ali, M. Shabir, M. Naz, Algebraic structures of soft sets associated with new operations, Computers and Mathematics with Applications 61 (9) (2011) 2647-2654.
- N. Çağman, S. Enginoğlu, Soft set theory and uni–int decision making, European Journal of Operational Research 207 (2) (2010) 848-855.
- S. Enginoğlu, B. Arslan, Intuitionistic fuzzy parameterized intuitionistic fuzzy soft matrices and their application in decision-making, Computational and Applied Mathematics 39 (4) (2020) 325.
- İ. Demir, N-soft mappings with application in medical diagnosis, Mathematical Methods in the Applied Sciences 44 (8) (2021) 7343-7358.
- G. Ali, M. Sarwar, M. Nabeel, Novel group decision-making method based on interval-valued m-polar fuzzy soft expert information, Neural Computing and Applications 35 (30) (2023) 22313-22340.
- H. Aktaş, N. Çağman, Soft sets and soft groups, Information Sciences 177 (13) (2007) 2726-2735.
- U. Acar, F. Koyuncu, F. Tanay, Soft sets and soft rings, Computers and Mathematics with Applications 59 (11) (2010) 3458-3463.
- N. Çağman, F. Çıtak, H. Aktaş, Soft int-group and its applications to group theory, Neural Computing and Applications 21 (1) (2011) 151-158.
- F. Çıtak, N. Çağman, Soft int-rings and its algebraic applications, Journal of Intelligent and Fuzzy Systems 28 (3) (2015) 1225-1233.
- A. Sezgin, N. Çağman, F. Çıtak, $\alpha$-inclusions applied to group theory via soft set and logic, Communications Faculty of Science University of Ankara Series A1 Mathematics and Statistics 68 (1) (2019) 334-352.
- C. Jana, M. Pal, F. Karaaslan, A. Sezgin, ($\alpha,\beta$)-soft intersectional rings and ideals with their applications, New Mathematics and Natural Computation 15 (2) (2019) 333-350.
- J. Ghosh, D. Mandal, T. K. Samanta, Soft prime and semiprime int-ideals of a ring, AIMS Mathematics 5 (1) (2020) 732-745.
- J. Ghosh, D. Mandal, T. K. Samanta, Soft maximal and irreducible int-ideals of a ring, New Mathematics and Natural Computation 16 (1) (2020) 37-52.
- J. Ghosh, D. Mandal, T. K. Samanta, The soft Jacobson radical of a commutative ring, Armenian Journal of Mathematics 13 (11) (2021) 1-9.
- R. Barzegar, S. B. Hosseini, N. Çağman, Second type nilpotent soft subgroups, Afrika Matematika 34 (1) (2023) 17.
- A. I. Alajlan, A. M. Alghamdi, Soft groups and characteristic soft subgroups, Symmetry 15 (7) (2023) 1450.
- İ. H. Kanat, F. Çıtak, On soft near-prime ideals and soft 1-absorbing prime ideals with applications, Journal of Mathematics 2025 (1) (2025) 1548808.
- A. Yassine, M. J. Nikmehr, R. Nikandish, On 1-absorbing prime ideals of commutative rings, Journal of Algebra and Its Applications 20 (10) (2021) 2150175.
- S. Koç, Ü. Tekir, E. Yıldız, On weakly 1-absorbing prime ideals, Ricerche di Matematica 72 (2) (2023) 723-738.
- N. Groenewald, 1-absorbing prime ideals and weakly 1-absorbing prime ideals in noncommutative rings, Sao Paulo Journal of Mathematical Sciences 17 (2) (2023) 871-887.
- N. Groenewald, 1-absorbing and weakly 1-absorbing prime submodules of a module over a noncommutative ring, Afrika Matematika 34 (3) (2023) 54.
- A. Badawi, Y. E. Celikel, On 1-absorbing primary ideals of commutative rings, Journal of Algebra and Its Applications 19 (6) (2020) 2050111.
- E. Yıldız, Ü. Tekir, S., Koç, On $\phi$-1-absorbing prime ideals, Beiträge zur Algebra und Geometrie 62 (4) (2021) 907–918.
- E. M. Bouba, M. Tamekkante, Ü. Tekir, S. Koç, Notes on 1-absorbing prime ideals, Proceedings of the Bulgarian Academy of Sciences 75 (5) (2022) 631-639.
- R. Coleman, L. Zwald, Primary ideals, (2020) hal-03040606.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory, Mathematical Logic, Set Theory, Lattices and Universal Algebra |
| Journal Section | Research Article |
| Authors | |
| Submission Date | February 2, 2026 |
| Acceptance Date | March 5, 2026 |
| Publication Date | March 30, 2026 |
| DOI | https://doi.org/10.28979/jarnas.1880226 |
| IZ | https://izlik.org/JA66FW24ZC |
| Published in Issue | Year 2026 Volume: 12 Issue: 1 |