Classes of $F$-hyperideals In A Krasner $F^{(m,n)}$-Hyperring

Document Type : Research Paper


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


‎Krasner $F^{(m,n)}$-hyperrings were introduced and  investigated by Farshi and Davvaz. In this paper, our purpose is to  define and characterize  three particular classes of $F$-hyperideals in a Krasner $F^{(m,n)}$-hyperring, namely prime $F$-hyperideals, maximal $F$-hyperideals and  primary $F$-hyperideals, which extend similar concepts of ring context. Furthermore, we  examine the relations between these structures. Then a number of major conclusions are given to explain the general framework of these structures.


[1] R. Ameri, M.M. Zahedi, Fuzzy subhypermodules over fuzzy hyperrings, in: Sixth International Congress on AHA, (1996) 1-14.
[2] R. Ameri, M.M. Zahedi, Hypergroup and join spaces induced by a fuzzy subset, Pure Math. Appl., vol. 8 (1997) 155-168.
[3] R. Ameri, M. Norouzi, Prime and primary hyperideals in Krasner (m; n)-hyperrings, European Journal Of Combinatorics, (2013) 379-390.
[4] M. Anbarloei, n-ary 2-absorbing and 2-absorbing primary hyperideals in Krasner (m; n)-hyperrings, Matematicki Vesnik, vol. 71, no. 3 (2019) 250-262.
[5] M. Anbarloei, Uni ng the prime and primary hyperideals under one frame in a Krasner (m; n)-hyperring, Comm. Algebra, vol. 49 (2021) 3432-3446.
[6] A. Asadi, R. Ameri, Direct limit of Krasner (m,n)-hyperrings, Journal of Sciences, vol. 31, no. 1 (2020) 75-83.
[7] G. Crombez, On (m; n)- rings, Abh. Math. Semin. Univ., Hamburg, vol. 37 (1972) 180-199.
[8] G. Crombez, J. Timm, On (m; n)-quotient rings, Abh. Math. Semin. Univ., Hamburg, vol. 37 (1972) 200-203.
[9] S. Corsini, Prolegomena of hypergroup theory, Second edition, Aviani editor, Italy, (1993).
[10] P. Corsini, Join spaces, power sets, fuzzy sets, in: Proc. Fifth Internat. Congress of Algebraic Hyperstructures and Application, 1993, Iasi, Romania, Hadronic Press, Palm Harbor, USA, (1994) 45-52.
[11] P. Corsini, V. Leoreanu, Join spaces associated with fuzzy sets, J. Combinatorics Inf. Syst. Sci., vol. 20, no. 1-4 (1995) 293-303.
[12] P. Corsini, I. Tofan, On fuzzy hypergroups, Pure Math. Appl., vol. 8 (1997) 29-37.
[13] P. Corsini, Fuzzy sets, join spaces and factor spaces, Pure Math. Appl., vol. 11, no. 3 (2000) 439-446.
[14] P. Corsini, V. Leoreanu, Fuzzy sets and join spaces associated with rough sets, Rendiconti di Circolo Matematico di Palermo, vol. 51 (2002) 527-536
[15] P. Corsini, V. Leoreanu, Applications of hyperstructure theory, Advances in Mathematics, Kluwer Academic Publishers, vol. 5 (2003).
[16] B. Davvaz, Fuzzy Hv-groups, Fuzzy Sets Syst., vol. 101 (1999) 191-195.
[17] B. Davvaz, Fuzzy Hv submodules, Fuzzy Sets Syst., vol. 117 (2001) 477-484.
[18] B. Davvaz, P. Corsini, Generalized fuzzy sub-hyperquasigroups of hyperquasigroups, Soft Comput., vol. 10, no. 11 (2006) 1109-1114.
[19] B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, Palm Harbor, USA, (2007).
[20] B. Davvaz, T. Vougiouklis, n-ary hypergroups, Iran. J. Sci. Technol., vol. 30, no. A2 (2006) 165-174.
[21] B. Davvaz, Fuzzy Krasner (m; n)-hyperrings, Computers and Mathematics with Applications, vol. 59 (2010) 3879-3891.
[22] W. Dorente, Untersuchungen uber einen verallgemeinerten Gruppenbegri , Math. Z., vol. 29 (1928) 1-19.
[23] B. Fahid, Z. Dongsheng, 2-Absorbing -primary ideals of commutative rings, Kyungpook Mathematical Journal, vol. 57 (2017) 193-198.
[24] M. Farshi, B. Davvaz, Krasner F(m;n)-hyperrings, Iranian Journal of Fuzzy Systems, vol. 11, no. 6 (2014) 67-88.
[25] K. Hila, K. Naka, B. Davvaz, On (k; n)-absorbing hyperideals in Krasner (m; n)-hyperrings, Quarterly Journal of Mathematics, vol. 69 (2018) 1035-1046.
[26] E. Kasner, An extension of the group concept (reported by L.G. Weld) Bull. Amer. Math. Soc., vol. 10 (1904) 290-291.
[27] Ath. Kehagias, L-fuzzy join and meet hyperoperations and the associated L-fuzzy hyperalgebras, Rendiconti di Circolo Matematico di Palermo, vol. 51 (2002) 503-526.
[28] Ath. Kehagias, An example of L-fuzzy join space, Rendiconti di Circolo Matematico di Palermo, vol. 52 (2003) 322-350.
[29] V. Leoreanu, Canonical n-ary hypergroups, Ital. J. Pure Appl. Math., vol. 24(2008).
[30] V. Leoreanu-Fotea, B. Davvaz, n-hypergroups and binary relations, European J. Combin., vol. 29 (2008) 1027-1218.
[31] V. Leoreanu-Fotea, B. Davvaz, Roughness in n-ary hypergroups, Inform. Sci., vol. 178 (2008) 4114-4124.
[32] V. Leoreanu-Fotea, B. Davvaz, Fuzzy hyperrings, Fuzzy Sets Syst., vol. 160 (2009) 2366-2378.
[33] W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets Syst. vol. 8 (1982) 133-139.
[34] X. Ma, J. Zhan, B. Davvaz, Applications of rough soft sets to Krasner (m; n)-hyperrings and corresponding decision making methods, Filomat, vol. 32 (2018) 6599-6614.
[35] F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandenaves, Stockholm, (1934) 45-49.
[36] S. Mirvakili, B. Davvaz, Relations on Krasner (m; n)-hyperrings, European J. Combin., vol. 31(2010) 790-802.
[37] S. Mirvakili, B. Davvaz, Constructions of (m; n)-hyperrings, Matematicki Vesnik, vol. 67 (1) (2015) 1-16.
[38] M. Norouzi, R.Ameri, V. Leoreanu-Fotea, Normal hyperideals in Krasner (m; n)-hyperrings, An. St. Univ. Ovidius Constanta, vol. 26, no. 3, (2018) 197-211.
[39] S. Omidi, B. Davvaz, Contribution to study special kinds of hyperideals in ordered semihyperrings , J. Taibah Univ. Sci., vol. 11 (2017) 1083-1094.
[40] S. Ostadhadi-Dehkordi, B. Davvaz, A Note on Isomorphism Theorems of Krasner (m; n)- hyperrings, Arabian Journal of Mathematics, vol. 5 (2016) 103-115.
[41] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., vol. 35 (1971) 512-517.
[42] K. Sera midis, A. Kehagias, M. Konstantinidou, The L-fuzzy Corsini join hyperoperation , Ital. J. Pure Appl. Math., vol. 12 (2002) 83-90.
[43] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press Inc., Florida, (1994).
[44] A. Yassine, M.J. Nikmehr, R. Nikandish, n-Ary k-absorbing hyperideals in krasner (m, n)-hyperrings, Afrika Matematika, (2022) 19-33.
[45] L. A. Zadeh, Fuzzy sets, Inform Control vol. 8(1965) 338-353.
[46] M. M. Zahedi, R. Ameri, On the prime, primary and maximal subhypermodules, Ital. J. Pure Appl. Math., vol. 5 (1999) 61-80.
[47] M. M. Zahedi, A. Hasankhani, F-polygroups I, J. Fuzzy Math., vol. 4, no. 3, (1996) 533-548.
[48] M. M. Zahedi, A. Hasankhani, F-polygroups II, Information Sciences, vol. 89, no. 3-4, (1996) 225-243.
[49] J. Zhan, B. Davvaz, K.P. Shum, Generalized fuzzy hyperideals of hyperrings, Computers and Mathematics with Applications, vol. 56 (2008) 1732-1740.