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Spectral hyperspaces of Krasner hyperrings

Amartya Goswami

TL;DR

Problem: spectral spaces in Krasner hyperrings have not been studied; the paper proves that the hyperspace $\mathcal{I}^+_R$ of proper hyperideals is spectral. Approach: employs an adapted method that avoids requiring a pre-existing compact-open basis, leveraging the algebraic-lattice structure of $\mathcal{I}_R$ to obtain spectrality and using the Alexander subbasis theorem to prove the quasi-compactness of $\mathcal{I}^+_R$. Sobriety is established by the existence of generic points for irreducible closed subsets, and $\mathcal{I}^+_R$ is shown to be an open subspace of $\mathcal{I}_R$. Significance: furnishes the first concrete spectral-space example in Krasner hyperring theory and strengthens the link between hyperring algebra and spectral topology.

Abstract

The purpose of this note is to prove that the hyperspaces of proper hyperideals of Krasner hyperrings are spectral.

Spectral hyperspaces of Krasner hyperrings

TL;DR

Problem: spectral spaces in Krasner hyperrings have not been studied; the paper proves that the hyperspace of proper hyperideals is spectral. Approach: employs an adapted method that avoids requiring a pre-existing compact-open basis, leveraging the algebraic-lattice structure of to obtain spectrality and using the Alexander subbasis theorem to prove the quasi-compactness of . Sobriety is established by the existence of generic points for irreducible closed subsets, and is shown to be an open subspace of . Significance: furnishes the first concrete spectral-space example in Krasner hyperring theory and strengthens the link between hyperring algebra and spectral topology.

Abstract

The purpose of this note is to prove that the hyperspaces of proper hyperideals of Krasner hyperrings are spectral.
Paper Structure (1 section, 4 theorems, 6 equations)

This paper contains 1 section, 4 theorems, 6 equations.

Table of Contents

  1. Introduction and Preliminaries

Key Result

Lemma 1.1

Let $R$ be a hyperring. If $\{I_{\lambda}\}_{\lambda \in \Lambda}$ is a nonempty family of hyperideals of $R,$ then is also a hyperideal of $R$, where each term of the sum is taken over a finite subset of the index set $\Lambda$.

Theorems & Definitions (6)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • proof
  • Theorem 1.4
  • proof