On hyperideals of Krasner hyperrings based on derived unitary rings

Document Type : Research Paper


1 Department of Pure Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran

2 Mathematics Department, Vali-e-Asr University of Rafsanjan, Rafsanjan,Iran

3 Department of Mathematics, Islamic Azad University, Kerman Branch, Kerman, Iran

4 Center for Information Technologies and Applied Mathematics, University of Nova Gorica, Slovenia


In this paper first, we introduce and analyze the strongly regular relations $\lambda^*_{e}$ and $\Lambda^*_{e}$ on a hyperring such that the derived quotient ring is unitary and unitary commutative, respectively. Next, we define and study the notion of $\lambda_e$-parts in a hyperring and  characterize the $\lambda_e$-parts in a $\lambda_e$-strong hyperring $R$. Finally, we introduce the notion of $\lambda_e$-closed hyperideal in a hyperring and study some of its fundamental properties in Krasner hyperrings.


[1] A. Adineh Zadeh, M. Norouzi, I. Cristea, The commutative quotient structure of m-idempotent hyperrings, An. St. Univ. Ovidius Constanta 28(1) (2020), 219-236.
[2] R. Ameri, M. Hamidi, H. Mohamadi, Hyperideals of (Finite) General Hyperrings, Mathematics Interdisciplinary Research, 6 (2021), 257{273.
[3] S. Breaz, C. Pelea, Multialgebras and term functions over the algebra of their nonvoid subsets, Mathematica (Cluj), 43(66) (2) (2001), 143{149.
[4] P. Corsini, Prolegomena of Hypergroup Theory, 2nd ed. Aviani Editore, Tricesimo, 1993.
[5] P. Corsini, V. Leoreanu, Applications of Hyperstructures Theory, Advanced in Mathematics, Kluwer Academic Publishers, 2003.
[6] G. Cupona, R. Madarasz, On poly-algebras, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat.Fak. Ser. Mat. 21 (2) (1991), 141{156.
[7] G. Cupona, R. Madarasz, Free poly-algebras, Univ. u Novom Sadu Zb. Rad. Prirod.Mat. Fak. Ser. Mat. 23 (2) (1993), 245{261.
[8] B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, USA, 2007.
[9] B. Davvaz, T. Vougiouklis, Commutative rings obtained from hyperrings (Hv-rings) with  -relations Comm. Algebra, 35 (2007), 3307-3320.
[10] M. De Salvo, G. Lo Faro, On the n-complete hypergroups, Discrete Math. 208/209 (1990), 177-188.
[11] D. Freni, A new characterization of the derived hypergroup via strongly regular equivalences, Comm. Algebra 30 (8) (2002), 3977-3989.
[12] P. Ghiasvand, S. Mirvakili, B. Davvaz, Boolean Rings Obtained from Hyperrings with 1;m Relations, Iranian Journal of Science and Technology, Transactions A: Science 41 (1) (2017), 69-79.
[13] M. Iranmanesh, M. Jafarpour, H. Aghabozorgi, J. Zhan, Classi cation of Krasner Hyper elds of Order 4, Acta. Math. Sin.-English Ser. 36 (2020), 889-902.
[14] M. Krasner, A class of hyperrings and hyper elds, Internat. J. Math. Math. Sci. 6 (2) (1983), 307-311.
[15] F. Marty, Sur une Generalization de la Notion de Groupe, 8th Congress Math. Scandinaves, Stockholm, Sweden, 45-49, 1934.
[16] C. G. Massouros, On the theory of hyperrings and hyper elds, Algebra i Logika 24 (1985), 728-742.
[17] S. Mirvakili, S. M. Anvariyeh, B. Davvaz, On  -relation and transitivity condition of  ,Comm. Algebra 36 (5) (2008), 1695-1703.
[18] S. Mirvakili, B. Davvaz, Applications of the  -relation to Krasner hyperrings, J. Algebra 362 (2012), 145-156.
[19] S. Mirvakili, B. Davvaz, Relationship between rings and hyperrings by using the notion of fundamental relations, Communications in Algebra 41 (1) (2013), 70-82.
[20] S. Mirvakili, P. Ghiasvand, B. Davvaz, Finitely generated rings obtained from hyperrings through the fundamental relations, Boletim da Sociedade Paranaense de Matematica 39 (1) (2021), 51-69.
[21] A. Nakassis, Expository and Survey Article: Recent Result in hyperring and Hyper eld Theory, Internet. J. Math and Math. Sci. 11 (2) (1988), 209- 220.
[22] M. Norouzi, I. Cristea, Fundamental relation on m-idempotent hyperrings, Open Math. 15 (2017), 1558-1567.
[23] D. M. Olson, V. K. Ward, A note on multiplicative hyperrings, Italian J. Pure Appl. Math. 1 (1997), 77-84.
[24] C. Pelea, On the fundamental relation of a multialgebra, Italian J. Pure Appl. Math. 10 (2001), 141-146.
[25] C. Pelea, I. Purdea, Multialgebras, universal algebras and identities, J. Aust. Math. Soc. 81 (2006), 121-139.
[26] C. Pelea, Hyperrings and  -relations. A general approach, J. Algebra 383 (1) (2013) 104{128.
[27] I.G. Rosenberg, An algebraic approach to hyperalgebras, in: Proceedings of 26th ISMVL, Santiago de Compostela, May 28{31, IEEE, 1996, 203-207.
[28] I.G. Rosenberg, Multiple-valued hyperstructures, in: Proceedings of 28th ISMVL, Fukuoka, May 27-29, IEEE, 1998, 326{333.
[29] R. Rota, Sugli iperanelli moltiplicativi Rend. Mat. 4 (1982) v.2 s.VII.
[30] T. Vougiouklis, Representations of hypergroups by hypermatrices, Rivista di Mat. Pure ed Appl. 2 (1987), 7-19.
[31] T. Vougiouklis, The fundamental relation in hyperrings. The general hyper elds, Proc. Forth Int. Congress on Algebraic hyperstructures and Applications, World Sci., 1991.
[32] T. Vougiouklis, Hyperstructures and their representation, Hadronic press, 115, Inc, Palm Harbor, USA, 1994.
Volume 11, Issue 3 - Serial Number 23
Special Issue dedicated to Prof. Mashaallah Mashinchi.
November 2022
Pages 33-56
  • Receive Date: 12 January 2022
  • Revise Date: 19 April 2022
  • Accept Date: 29 April 2022
  • First Publish Date: 10 May 2022