$\phi$-$(k,n)$-absorbing (primary) hyperideals in a Krasner $(m,n)$-hyperring

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin, Iran.

Abstract

Various expansions of prime hyperideals have been studied in a Krasner $(m,n)$-hyperring $R$. For instance, a proper hyperideal $Q$ of $R$ is called weakly $(k,n)$-absorbing (primary) provided that for $r_1^{kn-k+1} \in R$, $g(r_1^{kn-k+1}) \in Q-\{0\}$ implies that there are $(k-1)n-k+2$ of the $r_i^,$s whose $g$-product is in $Q$ $\Bigl ($ $g(r_1^{(k-1)n-k+2}) \in Q$ or a $g$-product of $(k-1)n-k+2$ of $r_i^,$s ,except $g(r_1^{(k-1)n-k+2})$, is in $\boldsymbol{ r}^{(m,n)}(Q)$ $\Bigr )$. In this paper, we aim to extend the notions to the concepts of $\phi$-$(k,n)$-absorbing and $\phi$-$(k,n)$-absorbing primary hyperideals. Assume that $\phi$ is a function from $ \mathcal{HI}(R)$ to $\mathcal{HI}(R) \cup \{\varnothing\}$ such that $\mathcal{HI}(R)$ is the set of hyperideals of $R$ and $k$ is a positive integer. We call a proper hyperideal $Q$ of $R$ a $\phi$-$(k,n)$-absorbing (primary) hyperideal if for $r_1^{kn-k+1} \in R$, $g(r_1^{kn-k+1}) \in Q-\phi(Q)$ implies that there are $(k-1)n-k+2$ of the $r_i^,$s whose $g$-product is in $Q$ $\Bigl ($ $g(r_1^{(k-1)n-k+2}) \in Q$ or a $g$-product of $(k-1)n-k+2$ of $r_i^,$s ,except $g(r_1^{(k-1)n-k+2})$, is in $\boldsymbol{ r}^{(m,n)}(Q)$ $\Bigr )$. Several properties and characterizations of them are presented.

Keywords

References

[1] R. Ameri and M. Norouzi, Prime and primary hyperideals in Krasner (m, n)-hyperrings, European J. Combin., 34 (2013) 379-390.
[2] R. Ameri, A. Kordi and S. Sarka-Mayerova, Multiplicative hyperring of fractions and coprime hyperideals, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 25 No. 1 (2017) 5-23.
[3] M. Anbarloei, n-ary 2-absorbing and 2-absorbing primary hyperideals in Krasner (m, n)-hyperrings, Mat. Vesn., 71 No. 3 (2019) 250-262.
[4] M. Anbarloei, Unifing the prime and primary hyperideals under one frame in a Krasner (m, n)-hyperring, Comm. Algebra, 49 (2021) 3432-3446.
[5] M. Anbarloei, A study on a generalization of the n-ary prime hyperideals in Krasner (m, n)-hyperrings, Afr. Mat., 33 (2021) 1021-1032.
[6] M. Anbarloei, Krasner (m, n)-hyperring of fractions, Jordan J. Math. Stat., 16 No. 1 (2023) 165-185.
[7] D. D. Anderson and M. Bataineh, Generalizations of prime ideals, Comm. Algebra, 36 (2008) 686-696.
[8] A. Asadi and R. Ameri, Direct limit of Krasner (m,n)-hyperrings, J. Sci., 31 No. 1 (2020) 75-83.
[9] A. Badawi, U. Tekir, E. A. Ugurlu, G. Ulucak and E. Y. Celikel, Generalizations of 2-absorbing primary ideals of commutative rings, Turkish J. Math., 40 No. 3 (2016) 703-717.
[10] S. Corsini, Prolegomena of Hypergroup Theory, Second edition, Aviani editor, Italy, 1993.
[11] S. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Advances in Mathematics, Vol. 5, Kluwer Academic Publishers, 2003.
[12] A. Y. Darani, Generalizations of primary ideals in commutative rings, Novi Sad J. Math., 42 No.1 (2012) 27-35.
[13] B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, Palm Harbor, USA, 2007.
[14] B. Davvaz and T. Vougiouklis, n-ary hypergroups, Iran. J. Sci. Technol., 30 No. A2 (2006) 165-174.
[15] B. Davvaz, Fuzzy Krasner (m, n)-hyperrings, Comput. Math. Appl., 59 (2010) 3879-3891.
[16] B. Davvaz, G. Ulucak and U. Tekir, Weakly (k, n)-absorbing (Primary) hyperideals of a Krasner (m, n)-hyperring, Hacet. J. Math. Stat., 52 No. 5 (2023) 1229-1238
[17] W. Dorente, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Math. Z., 29 (1928) 1-19.
[18] K. Hila, K. Naka and B. Davvaz, On (k, n)-absorbing hyperideals in Krasner (m, n)-hyperrings, Q. J. Math., 69 (2018) 1035-1046.
[19] A. Jaber, Properties of ϕ-δ-primary and 2-absorbing δ-primary ideals of commutative rings, Asian-Eur. J. Math., 13 No. 1 (2020), 2050026.
[20] A. Khaksari, ϕ-2-prime ideals, Int. J. Pure Appl. Math., 99 No. 1 (2015) 1-10.
[21] E. Kasner, An extension of the group concept (reported by L.G. Weld), Bull. Amer. Math. Soc., 10 (1904) 290-291.
[22] V. Leoreanu-Fotea, Canonical n-ary hypergroups, Ital. J. Pure Appl. Math., 24 (2008) 247-254.
[23] V. Leoreanu-Fotea and B. Davvaz, n-hypergroups and binary relations, European J. Combin., 29 (2008) 1027-1218.
[24] V. Leoreanu-Fotea and B. Davvaz, Roughness in n-ary hypergroups, Inform. Sci., 178 (2008) 4114-4124.
[25] X. Ma, J. Zhan and B. Davvaz, Applications of rough soft sets to Krasner (m, n)-hyperrings and corresponding decision making methods, Filomat, 32 (2018) 6599-6614.
[26] F. Marty, Sur une generalization de la notion de groupe, In 8th Congress Math. Scandenaves, Stockholm, 1934.
[27] S. Mirvakili and B. Davvaz, Relations on Krasner (m, n)-hyperrings, European J. Combin., 31 (2010) 790-802.
[28] S. Mirvakili and B. Davvaz, Constructions of (m, n)-hyperrings, Mat. Vesn., 67 No. 1 (2015) 1-16.
[29] M. Norouzi, R.Ameri and V. Leoreanu-Fotea, Normal hyperideals in Krasner (m, n)-hyperrings, An. St. Univ. Ovidius Constanta, 26 No. 3 (2018) 197-211.
[30] S. Omidi and B. Davvaz, Contribution to study special kinds of hyperideals in ordered semihyperrings, J. Taibah Univ. Sci., 11 (2017) 1083-1094.
[31] S. Ostadhadi-Dehkordi and B. Davvaz, A Note on isomorphism theorems of Krasner (m, n)-hyperrings, Arab. J. Math., 5 (2016) 103-115.
[32] T. Vougiouklis, Hyperstructures and their Representations, Hadronic Press Inc., Florida, 1994.
[33] M. M. Zahedi and R. Ameri, On the prime, primary and maximal subhypermodules, Ital. J. Pure Appl. Math., 5 (1999) 61-80.
[34] J. Zhan, B. Davvaz and K. P. Shum, Generalized fuzzy hyperideals of hyperrings, Comput. Math. Appl., 56 (2008) 1732-1740.
Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 16 May 2024

Files

Share

How to cite

Statistics