ESTONIAN ACADEMY
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eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
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proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Research article
δ-r-hyperideals and φ-δ-r-hyperideals of commutative Krasner hyperrings; pp. 96–103
PDF | https://doi.org/10.3176/proc.2023.1.09

Authors
Peng Xu, Melis Bolat, Elif Kaya, Serkan Onar, Bayram Ali Ersoy, Kostaq Hila
Abstract

This paper deals with an important class of multialgebras, called Krasner hyperrings. Our purpose is to define the expansion of r-hyperideals and to extend this concept to φ-δ-r-hyperideal in commutative Krasner hyperrings with nonzero identity. δ-r-hyperideals of commutative Krasner hyperrings are studied. Some properties of φ-δ-r-hyperideals are investigated and several examples are provided.

References

1. Asokkumar, A. and Velrajan, M. Von Neumann regularity on Krasner hyperring. In Algebra, Graph Theory and Their Applica- tions (Tamizh Chelvam, T., Somasundaram, S. and Kala, R., eds). Narosa Publishing House, New Delhi, 2010. 

2. Al Tahan, M. and Davvaz, B. On the existence of hyperrings associated to arithmetic functions. J. Number Theory, 2017, 174, 136–149.
https://doi.org/10.1016/j.jnt.2016.10.017

3. Al Tahan, M. and Davvaz, B. Strongly regular relations of arithmetic functions. J. Number Theory, 2018, 187, 391–402.
https://doi.org/10.1016/j.jnt.2017.11.006

4. Anderson, D. D. and Bataineh, M. Generalizations of prime ideals. Commun. Algebra, 2008, 36(2), 686–696.
https://doi.org/10.1080/00927870701724177

5. Corsini, P. Prolegomena of Hypergroup Theory. Aviani, Tricesimo, 1993.

6. Corsini, P. and Leoreanu, V. Applications of Hyperstructure Theory. (ADMA 5). Kluwer Academic Publishers, Dordrecht, 2003.
https://doi.org/10.1007/978-1-4757-3714-1

7. Darani, A. Y. Generalizations of primary ideals in commutative rings. Novi Sad J. Math., 2012, 42(1), 27–35.

8. Davvaz, B. Polygroup Theory and Related Systems. World Scientific, Hackensack, 2013.
https://doi.org/10.1142/8593

9. Davvaz, B. and Leoreanu-Fotea, V. Hyperring Theory and Applications. International Academic Press, Palm Harbor, 2007. 

10. Davvaz, B. and Vougiouklis, T. A Walk Through Weak Hyperstructures: Hv-structures. WSPC, Hackensack, 2019. 
https://doi.org/10.1142/11229

11. Davvaz, B. and Leoreanu-Fotea, V. Hypergroup Theory. WSPC, Hackensack, 2022.
https://doi.org/10.1142/12645

12. Davvaz, B. and Salasi, A. A realization of hyperrings. Commun. Algebra, 2006, 34(12), 4389–4400.
https://doi.org/10.1080/00927870600938316

13. Zhao, D. δ-primary ideals of commutative rings. Kyungpook Math. J., 2001, 41(1), 17–22. 

14. Guan, H., Kaya, E., Bolat, M., Onar, S., Ersoy, B. A. and Hila, K. φ-δ-primary hyperideals in Krasner hyperrings. Math. Probl. Eng., 2022, 2022, 1192684. 

15. Huckaba, J. A. Commutative Rings With Zero Divisors. Marcel Dekker, Inc., New York, 1988.

16. Jaber, A. Properties of φ-δ-primary and 2-absorbing δ-primary ideals of commutative rings. Asian-European J. Math., 2020, 13(1), 2050026.
https://doi.org/10.1142/S1793557120500266

17. Krasner, M. A class of hyperrings and hyperfields. Int. J. Math. Math. Sci., 1983, 6(2), 307–312.
https://doi.org/10.1155/S0161171283000265

18. Krasner, M. Approximation des corps valués complets de charactéristique ≠ 0 par ceux de caractéristique 0. In Colloque d’algebre superieure, Bruxelles 19–22 December 1956. Center Belge de Recherches Mathematiques, 1957, 129–206.

19. Marty, F. Sur une generalization de la notion de groupe. In Proceedings of the 8th Scandinavian Congress of Mathematicians, Stockholm, Sweden, 14–18 August, 1934. Ohlssons, 45–49.

20. Massouros, C. and Massouros, G. An overview of the foundations of the hypergroup theory. Mathematics, 2021, 9(9), 1014. 
https://doi.org/10.3390/math9091014

21. Massouros, C. On the theory of hyperrings and hyperfields. Algebra Log., 1985, 24, 728–742. 
https://doi.org/10.1007/BF01978850

22. Mittas, J. Hypergroupes canoniques. Math. Balkanica, 1972, 2, 165–179.

23. Massouros, G. and Massouros, C. Hypercompositional algebra, computer science and geometry. Mathematics, 2020, 8(8), 1338. 
https://doi.org/10.3390/math8081338

24. Mohamadian, R. r-ideals in commutative rings, Turkish J. Math., 2015, 39(5), 733–749.
https://doi.org/10.3906/mat-1503-35

25. Nakassis, A. Recent results in hyperring and hyperfield theory. Int. J. Math. Math. Sci., 1988, 11(2), 209–220.
https://doi.org/10.1155/S0161171288000250

26. Omidi, S., Davvaz, B. and Zhan, J. Some properties of n-hyperideals in commutative hyperrings, J. Algebraic Hyperstruct. Logical Algebras, 2020, 1(2), 23–30.
https://doi.org/10.29252/hatef.jahla.1.2.3

27. Ozel Ay, E. δ-primary hyperideals on commutative hyperrings, Int. J. Math. Math. Sci., 2017, 2017, 5428160.
https://doi.org/10.1155/2017/5428160

28. Stefanescu, M. 2006. Constructions of hyperfields and hyperrings. Stud. Cercet. Stiint. Ser. Mat. Univ. Bacau, 16, 563–571. 

29. Ugurlu, E. A. Generalizations of r-ideals of commutative rings. J. Interdisciplinary Math., 2021, 24(8), 2283–2293.
https://doi.org/10.1080/09720502.2021.1876294

30. Vougiouklis, T. Hyperstructures and Their Representations. Hadronic Press, Palm Harbor, 1994.

31. Vougiouklis, T. The fundamental relation in hyperrings. The general hyperfield. In Proceedings of the 4th International Congress on Algebraic Hyperstructures and Applications (AHA), 1990. World Scientific, 1991, 203–211.
https://doi.org/10.1142/9789814539555

32. Xu, P., Bolat, M., Kaya, E., Onar, S., Ersoy, B. A. and Hila, K. r-hyperideals and generalizations of r-hyperideals in Krasner hyperrings. Math. Probl. Eng., 2022, 2022, 7862425.

33. Xu, P., Bolat, M., Kaya, E., Onar, S., Ersoy, B. A. and Hila, K. δ-r-hyperideals and φ-δ-r-hyperideals of commutative Krasner hyperrings. 2021. 
https://arxiv.org/abs/2112.05513

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