Integral closures, Primary Hyperideals and Hypervaluation Hyperideals of Kranser Hyperrings

TL;DR

This paper introduces the concepts of integral closure and hypervaluation hyperideals in Krasner hyperrings, exploring their properties and relationships, and demonstrating how hypervaluations determine the integral closure of hyperideals.

Contribution

It defines the integral closure of hyperrings and hyperideals, introduces hypervaluation hyperideals, and studies their interrelations within Krasner hyperrings, advancing hyperring theory.

Findings

Integral closure of hyperideals is characterized by hypervaluations.

Hypervaluation hyperideals are linked to the integral closure.

The structure of Krasner hyperrings influences hyperideal properties.

Abstract

In this paper, the notions of integral closure of hyperrings and hyperideals in a Krasner hyperring $(R, +, \cdot)$ are defined and some basics properties of them are studied. We define also the notion of hypervaluation hyperideals and then a relations between hypervaluations, integral closure of hyperideals and primary hyperideals are studied. In fact it is shown that the integral closure of a hyperideal is determined by the hypervaluation Krasner hyperrings.