We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK = AK/K of a global field K. After promoting F1 to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G ⊂ R× is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G ∪ {0} is a subfi...

This paper considers the relations among L -fuzzy sets, rough sets and hyperring theory. Based on a complete residuated lattice, the concept of (invertible) L-fuzzy hyperideals of a hyperring is introduced and some related properties are presented. The notions of lower and upper L-fuzzy rough approximation operators with respect to an L-fuzzy hyperideal are provided and some significant propert...

In this paper, our aim is to generalize and extend the Hopkins-Levitzki theorem from noncommutative rings Krasner hyperring. Also, we prove that a hyperring R Noetherian if only it satisfies ascending chain conditions of prime hyperideals.

A class of hyperrings known as divisible will be studied in this paper. It presented each element hyperring is a element. Also shows the relationship between Jacobsen Radical, and set invertible elements gets some results, linked these results with hyperring. After going through concept hypermodule that by Sopon Boriboon Sajee Pianskool 2017, later 2022 Hashem Bordbar, Irina Cristea, related to...

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 hyperringswere introduced by Davvaz and Vougiouklis (2007). Recently, another strongly regular relation named θ on hyperrings has been studied by Ameri and Norouzi (2013). Ameri...

Let G be a group (monoid) with identity e and R commutative Krasner hyperring. In this paper, we introduce the concepts of graded absorbing hyperideals hyperring such as, 2-absorbing hyperideals, n-absorbing subhypermodules. Some basic properties these structures characterizations homogeneous components are proved.

This survey article presents some recent results in the theory of hyperfields and hyperrings, algebraic structures for which the "sum" of two elements is a subset of the structure. The results in this paper show that these structures .cannot always be embedded in the decomposition of an ordinary structure (ring or field) in equivalence classes and that the structural results for hyperfields and...

In this paper we introduce and analyze the notion of fuzzy hyperideal of a fuzzy hyperring. Then we investigate some properties based on homomorphisms between fuzzy hyperrings. Also, we try to extend the notion of cuts in fuzzy structures for fuzzy hyperrings. We study the quotient structure of a fuzzy hyperring and also we define fuzzy hypercongruences. Then we investigate some connections bet...

In this paper, we introduce new expansion classes, namely weakly $ (k,n) $-absorbing hyperideals and primary of a Krasner (m,n) $-hyperring, including hyperideal hyperideal. Therefore, give generalizations Also, examine the relations between classical explore some ways to connect them. Additionally, main results examples are given explain structures these concepts. Finally, study version Nakaya...