On $(t,n)$-absorbing $\delta$-semiprimary hyperideals

Articles in Press

Document Type : Research Paper

Author

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

Abstract

The $\delta$-primary hyperideal is a concept unifing the $n$-ary prime and $n$-ary primary hyperideals under one frame where $\delta$ is a function which assigns to each hyperideal $Q$ of a hyperring $G$ a hyperideal $\delta(Q)$ of the same hyperring with specific properties. In this paper, for a commutative Krasner $(m,n)$-hyperring $(G,h,k)$ with scalar identity $1$, we aim to introduce and study the notion of $(t,n)$-absorbing $\delta$-semiprimary hyperideals which is a more general structure than $\delta$-primary hyperideals. We say that a proper hyperideal $Q$ of $G$ is an $(t,n)$-absorbing $\delta$-semiprimary hyperideal if whenever $k(a_1^{tn-t+1}) \in Q$ for $a_1^{tn-t+1} \in G$, then there exist $(t-1)n-t+2$ of the $a_i^,$s whose $k$-product is in $\delta(Q)$. Furthermore, we extend the concept to weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideals. Several properties and characterizations of these classes of hyperideals are determined. In particular, after defining srongly weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideals, we present the condition in which a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal is srongly. Moreover, we show that $k(Q^{(tn-t+1)})=0$ where the weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal $Q$ is not $(t,n)$-absorbing $\delta$-semiprimary. We give a type of Nakayama$^,$s Lemma on a commutative Krasner $(m,n)$-hyperring. Also, we investigate the stability of the concepts under intersection, homomorphism and cartesian product of hyperrings.

Keywords

References

[1] J. Azami, A short note on prime submodules, Alg. Struc. Appl., 5 No. 1 (2018) 41-49.
[2] R. Ameri and M. Norouzi, Prime and primary hyperideals in Krasner (m, n)-hyperrings, European J. Combin., 34 No. 2 (2013) 379-390.
[3] M. Anbarloei, Unifing the prime and primary hyperideals under one frame in a Krasner (m, n)-hyperring, Comm. Algebra, 49 No. 8 (2021) 3432-3446.
[4] S. M. Anvariyeh, S. Mirvakili and B. Davvaz, Fundamental relation on (m, n)-hypermodules over (m, n)-hyperrings, Ars Combin., 94 (2010) 273-288.
[5] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75 No. 3 (2007) 417-429.
[6] E. Y. Celikel, 2-absorbing δ-semiprimary ideals of commutative rings, Kyungpook Math. J., 61 No. 4 (2021) 711-725.
[7] B. Davvaz and T. Vougiouklis, n-ary hypergroups, Iran. J. Sci. Technol., 30 No. 2 (2006) 165-174.
[8] B. Davvaz, Fuzzy Krasner (m, n)-hyperrings, Comput. Math. Appl., 59 (2010) 3879-3891.
[9] 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.
[10] W. Dorente, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Math. Z., 29 (1928) 1-19.
[11] F. Farshadifar, 2-absorbing I-prime and 2-absorbing I-second submodules, Alg. Struc. Appl., 6 No. 2 (2019) 47-55.
[12] K. Hila, K. Naka and B. Davvaz, On (k, n)-absorbing hyperideals in Krasner (m, n)-hyperrings, Q. J. Math., 69 No. 3 (2018) 1035-1046.
[13] E. Kasner, An extension of the group concept (reported by L.G. Weld), Bull. Amer. Math. Soc., 10 (1904) 290-291.
[14] V. Leoreanu, Canonical n-ary hypergroups, Ital. J. Pure Appl. Math., 24 (2008).
[15] V. Leoreanu-Fotea and B. Davvaz, n-hypergroups and binary relations, European J. Combin., 29 No. 5 (2008) 1027-1218.
[16] V. Leoreanu-Fotea and B. Davvaz, Roughness in n-ary hypergroups, Inform. Sci., 178 No. 21 (2008) 4114-4124.
[17] X. Ma, J. Zhan and B. Davvaz, Applications of rough soft sets to Krasner (m, n)-hyperrings and corresponding decision making methods, Filomat, 32 No. 19 (2018) 6599-6614.
[18] S. Mirvakili and B. Davvaz, Relations on Krasner (m, n)-hyperrings, European J. Combin., 31 No. 3 (2010) 790-802.
[19] M. H. Naderi, Uniformly classical quasi-primary submodules, Alg. Struc. Appl., 2 No. 1 (2015) 35-47.
[20] M. J. Nikmehr, R. Nikandish, A. Yassine, K. Hila and S. Hoskova-Mayerova, A generalization of n-ary prime subhypermodule, An. oSt. Univ. Ovidius Constanota, 32 No. 3 (2024) 103-123.
[21] S. Ostadhadi-Dehkordi and B. Davvaz, A note on Isomorphism Theorems of Krasner (m, n)- hyperring, Arab. J. Math., 5 No. 2 (2016) 103-115.
[22] M. M. Zahedi and R. Ameri, On the prime, primary and maximal subhypermodules, Ital. J. Pure Appl. Math., 5 (1999) 61-80.
[23] M. Zolfaghari and M.H Moslemi Koopaei, Some remarks on generalizations of classical prime submodules, Alg. Struc. Appl., 6 No. 2 (2019) 67-80.
Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 09 February 2026

Files

Share

How to cite

Statistics