δ-Primary Hyperideals on Commutative Hyperrings (original) (raw)

On Prime and Primary Hyperideals of a Multiplicative Hyperring

Annals of the Alexandru Ioan Cuza University - Mathematics, 2012

Procesi and Rota introduced and studied in brief the prime hyperideals of multiplicative hyperrings. Here we intend to investigate extensively the prime and primary hyperideals of multiplicative hyperrings with absorbing zero. Defining the radical of a hyperideal I of a multiplicative hyperring with absorbing zero, as the intersection of all prime ideals containing I, we obtain a generalized version of Krull's theorem regarding the structure of the radical of a particular class of hyperideals, called C-ideals of a multiplicative hyperring. In the last section of this paper, we describe the prime hyperideals, primary hyperideals and C-ideals of a multiplicative hyperring ZA over the ring of integers Z, induced by any A ∈ P * (Z).

φ-δ-Primary Hyperideals in Krasner Hyperrings

2021

In this paper, we study commutative Krasner hyperring with nonzero identity. φ-prime, φ-primary and φ-δ-primary hyperideals are introduced. We intend to extend the concept of δ-primary hyperideals to φ-δprimary hyperideals. We give some characterizations of hyperideals to classify them. We denote the set of all hyperideals of R by L(R) (all proper hyperideals of R by L∗(R)). Let φ be a reduction function such that φ : L(R) → L(R)∪{∅} and δ be an expansion function such that δ : L(R) → L(R). N be a proper hyperideal of R. N is called φ-δ-primary hyperideal of R if a ◦ b ∈ N− φ(N), then a ∈ N or b ∈ δ(N), for some a, b ∈ R. We discuss the relation between φ-δ-primary hyperideal and other hyperideals.

A note on weakly prime hyperideals and (1, n) hyperideals on multiplicative hyperrings

Thermal Science

In this article we give the definition of weakly prime hyperideals over multiplicative hyperring. We provide important results showing the relations between prime hyperideals and weakly prime hyperideals. Then we give definition of weakly n-hyperideal and weakly (1,n) hyperideal over multiplicative hyperrings. Then we investigate of their properties and give some examples. Hyperstructures are very important topic because it satisfy to study multidisiplinary. Especially in chemistry, coding theory, geometry, artificial intelligence, etc.

A note on “New fundamental relation of hyperrings”

European Journal of Combinatorics, 2014

In the theory of hyperrings, fundamental relations make a connection between hyperrings and ordinary rings. Commutative fundamental rings and the fundamental relation α * which is the smallest strongly regular relation in hyperrings were introduced by Davvaz and Vougiouklis . Recently, another strongly regular relation named θ * on hyperrings has been studied by . Ameri and Norouzi proved that θ * is the smallest strongly regular relation such that R/θ * is a commutative ring. In this paper, we show that θ * ̸ = α * and θ * is not the smallest strongly regular relation. Moreover, we show that some results of Ameri and Norouzi do not hold.

On S-prime hyperideals in multiplicative hyperrings

Journal of Algebraic Hyperstructures and Logical Algebras, 2021

Let R be a multiplicative hyperring and S ⊆ R be a multiplicatively closed subset of R. In this paper, we introduce and study the concept of S-prime hyperideals which is a generalization of prime hyperideals. Some properties of S-prime hyperideals in multiplicative hyperring are presented. Then we investigate the behaviour of S-prime hyperideals under homomorphism hyperrings, in factor hyperrings, Cartesian products of hyperrings, and the fundamental relation in the context of multiplicative hyperring.

r -Hyperideals and Generalizations of r -Hyperideals in Krasner Hyperrings

Mathematical Problems in Engineering

This paper deals with Krasner hyperrings as an important class of algebraic hyperstructures. We investigate some properties of r -hyperideals in commutative Krasner hyperrings. Some properties of p r -hyperideals are also studied. The relation between prime hyperideals and r -hyperideals is investigated. We show that the image and the inverse image of an r -hyperideal are also an r -hyperideal. We also introduce a generalization of r -hyperideals, and we prove some properties of them.

New fundamental relation of hyperrings

European Journal of Combinatorics, 2013

We prove that θ * as previously defined [Int. J. Contemp. Math. Sci. 5 (2010) 721] is the smallest equivalence relation such that the quotient structure R/θ * is a commutative fundamental ring. We also investigate some properties with respect to the commutative fundamental relation θ * on a hyperring R.

A new characterization of fundamental relation on hyperrings

In this note we introduce a new equivalence relation θ * on a (semi-)hyperring R and we show that it is strongly regular. Also we prove that, R/θ * , the equivalence class of this equivalence relation under usual operations consists a commutative (semi-)ring. Finally we introduce the notion of θ-parts of hyperrings and investigate the important properties of them.

Multiplicative hyperring of fractions and coprime hyperideals

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2017

In this paper we will introduce the notion of coprime hyperideals in multiplicative hyperrings and we will show some properties of them. Then we introduce the notion of hyperring of fractions generated by a multiplicative hyperring and then we will show some properties of them.

Regular Multiplicative Hyperrings

European Journal of Pure and Applied Mathematics, 2016

We introduce and study regular multiplicative hyperrings, as a generalization of classical rings. Also, we use the fundamental relation ? on a given regular multiplicative hyperring R and prove that the fundamental ring R=R/? of R is a regular ring. Finally, we investigate the algebraic properties of M(R), the regular hyperideal of R, generated by all elements of R such that its generated hyperideals is regular.