Superring of Polynomials over a Hyperring

Abstract

A Krasner hyperring (for short, a hyperring) is a generalization of a ring such that the addition is multivalued and the multiplication is as usual single valued and satisfies the usual ring properties. One of the important subjects in the theory of hyperrings is the study of polynomials over a hyperring. Recently, polynomials over hyperrings have been studied by Davvaz and Musavi, and they proved that polynomials over a hyperring constitute an additive-multiplicative hyperring that is a hyperstructure in which both addition and multiplication are multivalued and multiplication is distributive with respect to the addition. In this paper, we first show that the polynomials over a hyperring is not an additive-multiplicative hyperring, since the multiplication is not distributive with respect to addition; then, we study hyperideals of polynomials, such as prime and maximal hyperideals and prove that every principal hyperideal generated by an irreducible polynomial is maximal and Hilbert’s basis theorem holds for polynomials over a hyperring.

Hyperhomographies on Krasner Hyperfields

“…In the last few years, researchers in the hypercompositional structure theory have investigated, principally from a theoretical point of view, all types of hyperrings: general hyperrings [20], multiplicative hyperrings [15], additive hyperrings [21], superrings [22], but till now, only the Krasner hyperrings have found interesting and useful applications in number theory, algebraic geometry, scheme theory, as mentioned in the introductory part of this article. Here the authors continue the study on the research topic started in [1] about elliptic hypercurves defined on quotient Krasner hyperfield, with applications in cryptography [8].…”

Regular Parameter Elements and Regular Local Hyperrings

“…There are some other types of hyperrings: if the multiplication is a hyperoperation and the addition is a binary operation, we talk about the multiplicative hyperrings, defined by Rota [4]. If both, the addition and the multiplication, are hyperoperations with the additive part being a canonical hypergroup, then we have superrings [5], which were introduced by Mittas in 1973 [6]. Until now, the most well known and studied type of hyperrings is the Krasner hyperring, that has a plentitude of applications in algebraic geometry [7,8], tropical geometry [9], theory of matroids [10], schemes theory [11], algebraic hypercurves [12,13], hypermomographies [14].…”

Derived Hyperstructures from Hyperconics

“…M. Krasner introduced the concept of the hyperfield and hyperring in Algebra [20,21]. The theory which was developed for the hyperrings is generalizing and extending the ring theory [22][23][24][25]. There are different types of hyperrings [22,25,26].…”

“…The theory which was developed for the hyperrings is generalizing and extending the ring theory [22][23][24][25]. There are different types of hyperrings [22,25,26]. In the most general case a triplet (R, +, •) is a hyperring if (R, +) is a hypergroup, (R, •) is a semihypergroup and the multiplication is bilaterally distributive with regards to the addition [3].…”