Hyperideals of (Finite) General Hyperrings

Document Type : Original Scientific Paper


1 School of Mathematics, Statistic and Computer Sciences, University of Tehran, Tehran, I. R. Iran

2 Department of Mathematics, Payame Noor University, Tehran, I. R. Iran


A general hyperring is an algebraic hypercompositional system (R,+,·) with two hyperoperations ”+" and ” · ”, such that for all x,y ∈ R, x + y and x · y are non-empty subsets of R, and R satisfies the axioms similar to a ring. We introduce and study hyperideals of a general hyperring. In this regards, we construct a connection between classical rings and general hyperrings, specifically, we extend a ring to a general hyperring in nontrivial way. Moreover, a way to construct a general hyperring from set are given. Also, we concentrate on an important class of general hyperrings, which is called Krasner hyperrings, and discuss on their hyperideals. Finally, the set of all hyperideals of a finite general (resp. Krasner) hyperring are considered and its hyperideals are investigated.


[1] R. Ameri and I. G. Rosenberg, Congruences of multialgebras, J. Multiple Valued Log. Soft Comput. 15 (5-6) (2009) 525 − 536.
[2] R. Ameri and M. M. Zahedi, Hyperalgebraic systems, Italian J. Pure Appl. Math. 6 (1999) 21 − 32.
[3] R. Ameri, M. Hamidi and A. A. Tavakoli, Boolean rings based on multirings, J. Sci. I. R. Iran 32 (2) (2021) 159 − 168.
[4] H. Bordbara and I. Cristea, Height of prime hyperideals in Krasner hyperrings, Filomat 31 (19) (2017) 6153 − 6163.
[5] P. Corsini, Prolegomena of Hypergroup Theory, 2nd ed., Aviani Editor, Tricesimo, Italy, (1993).
[6] J. Chvalina, S. H. Mayerova and A. D. Nezhad, General actions of hyperstruc tures and some applications, Analele Stiintifice ale Univ. Ovidius Constanta, Ser. Mat. 21 (1) (2013) 59 − 82.
[7] P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academic Publishers, Dordrecht, 2002.
[8] B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, USA, 2007.
[9] M. Krasner, Approximation des corps values complets de caracteristique p, p > 0, par ceux de caracter- istique zero, Colloque d’Algebre Superieure (Bruxelles, Decembre 1956), CBRM, Bruxelles, 1957.
[10] F. Marty, Sur une generalization de la notion de groupe, 8th Congres Math. Scandinaves, Stockholm, Sweden (1934) 45 − 49.
[11] S. Omidi and B. Davvaz, Contribution to study special kinds of hyperideals in ordered semihyperrings, J. Taibah Univ. Sci. 11 (2017) 1083 − 1094.
[12] R. Rota, Sugli iperanelli moltiplicativi, Rend. Di Mat. Series VII 2 (4) (1982) 711 − 724.
[13] T. Vougiouklis, The fundamental relation in hyperrings, The general hypereld, Proc. Fourth Int. Congress on Algebraic Hyperstructures and Applications (AHA 1990), World Scientic (1991) 203 − 211.