On Hyperideals of Multiplicative Hyperrings
Abstract
Let R be a commutative multiplicative hyperring. In this paper, we introduce and study the concepts of n-hyperideal and
δ-n-hyperideal of R which are generalization of n-ideals and
δ-n-ideals of the in a commutative ring. An element a is called a nilpotent element of R if there exists a positive integer n such that 0∈a^n. A hyperideal I (I ≠R) of R is called an n- hyperideal of R if for all a,b∈R, a*b⊆I and a is non-nilpotent element implies that b∈I [15]. Also, I is called a
δ-n-hyperideal if for all a,b∈R, a*b⊆I then either a is nilpotent or b∈δ(I) , where
δ is an expansion function over the set of all hyperideals of a multiplicative hyperring. In addition, we give the definition of zd-hyperideal. Some properties of n-hyperideals,
δ-n-hyperideals and zd-hyperideals of the hyperring R are presented. Finally, the relations between these notions are investigated.
δ-n-hyperideal of R which are generalization of n-ideals and
δ-n-ideals of the in a commutative ring. An element a is called a nilpotent element of R if there exists a positive integer n such that 0∈a^n. A hyperideal I (I ≠R) of R is called an n- hyperideal of R if for all a,b∈R, a*b⊆I and a is non-nilpotent element implies that b∈I [15]. Also, I is called a
δ-n-hyperideal if for all a,b∈R, a*b⊆I then either a is nilpotent or b∈δ(I) , where
δ is an expansion function over the set of all hyperideals of a multiplicative hyperring. In addition, we give the definition of zd-hyperideal. Some properties of n-hyperideals,
δ-n-hyperideals and zd-hyperideals of the hyperring R are presented. Finally, the relations between these notions are investigated.
Keywords
Multiplicative hyperring, n-hyperideal, δ-n-hyperideal.
Project Number
(21/124/03/01)
References
- [1] Marty F., Sur une generalization de la notion de groupe, 8. Congress Math. Scandenaves, Stockholm, (1934) 45-49.
- [2] Corsini S., Leoreanu V., Applications of Hyperstructure Theory, Advances in Mathematics, Kluwer Academic Publishers, (2003).
- [3] Corsini S., Prolegomena of Hypergroup Theory, Aviani Editore, Italy, (1993).
- [4] Davvaz B., Leoreanu-Fotea V., Hyperring Theory and Applications. Palm Harbor, USA: International Academic Press (2007).
- [5] Rota R., Sugli iperanelli moltiplicativi, Rend. Di. Math., Series VII, 4 (1982) 711-724.
- [6] Ameri R., Eyvazi M., Hoskova-Mayerova S., Superring of Polynomials over a Hyperring, Mathematics, 7 (10) (2019) 1-15.
- [7] Davvaz B., Salasi A., A Realization of Hyperrings, Communications in Algebra, 34(2007) 4389-4400.
- [8] Ameri R., Eyvazi M., Hoskova-Mayerova S., Multiplicative Hyperring of Fractions and Coprime Hyperideals, An. St. Univ. Ovidius Constant, 25(1) (2017) 5-23.
- [9] Bayrak D., Yamak S., A Note on The Lattice of Fuzzy Hyperideals of a Hyperring, Afrika Matematika, 28(2017) 1185-1192.
- [10] Procesi R., Rota R., On Some Classes of Hyperstructures, Discrete Mathematics, 208/209 (1999) 485-497.