Superring of Polynomials over a Hyperring

Reza Ameri,Sarka Hoskova-Mayerova,Mansour Eyvazi

doi:10.3390/math7100902

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.

Highlights

A well established branch of classical algebraic theory is the theory of algebraic hyperstructures respectively hyperalgebraic system
A recent paper of Asadi and Ameri deals with categorical connection between categories (m, n)-hyperrings and (m, n)-rings via the fundamental relation [10]
Basis theorem holds for a Krasner hyperring R that is, if R is a Noetherian Krasner hyperring, so is the superring R[ x ]

Summary

Introduction

A well established branch of classical algebraic theory is the theory of algebraic hyperstructures respectively hyperalgebraic system. Hyperfield extension is one of the important topics in the theory of algebraic hyperstructures, which can be considered as a development of the classical field theory, but it is an important tool to study non-commutative geometry and algebraic geometry [25]. Contrary to polynomials over a ring (or a field) in classical algebra, the behaviour of polynomials over a hyperring or hyperfield is completely different and much more complicated, since the product of two polynomials is a polynomial, but it is a set of polynomials In this regard, we show that, for polynomials over a hyperring even over a hyperfield [30], the product is not distributive with respect to addition (Theorem 3.7); it has a weak distributive property, and it constitutes a hyperring, which is called a superring. Basis theorem holds for a Krasner hyperring R that is, if R is a Noetherian Krasner hyperring, so is the superring R[ x ]

Preliminaries

Polynomials over Krasner Hyperrings

Conclusions