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Spectral topologies on skew braces

Themba Dube, Amartya Goswami

TL;DR

This work develops a spectral-topological framework for skew braces by introducing a new notion of prime ideals and constructing a Grothendieck-style topology on $\mathrm{Spec}(A)$. It analyzes irreducible closed subsets, generic points, and Noetherian conditions, and shows how skew-brace homomorphisms induce continuous maps between spectra, with a detailed adjunction between ideals and closed sets. A central result is that $\mathrm{Spec}(\mathrm{Idl}(A))$ is a spectral space, linking lattice-theoretic and topological properties. The paper also clarifies the roles of the radical and nilradical in this setting and outlines several open questions for further study.

Abstract

Using a new definition of a prime ideal of a skew brace A, on set Spec A of prime ideals of A we endow a spectral topology (in the sense of Grothendieck). We characterize irreducible closed subsets of Spec A and prove every irreducible closed subset of the space has a unique generic point. We give a sufficient condition for the space to be Noetherian. We study continuous maps between such spaces, and finally, we prove that Spec(Idl A) is a spectral space, where Idl A is the set of all ideals of A.

Spectral topologies on skew braces

TL;DR

This work develops a spectral-topological framework for skew braces by introducing a new notion of prime ideals and constructing a Grothendieck-style topology on . It analyzes irreducible closed subsets, generic points, and Noetherian conditions, and shows how skew-brace homomorphisms induce continuous maps between spectra, with a detailed adjunction between ideals and closed sets. A central result is that is a spectral space, linking lattice-theoretic and topological properties. The paper also clarifies the roles of the radical and nilradical in this setting and outlines several open questions for further study.

Abstract

Using a new definition of a prime ideal of a skew brace A, on set Spec A of prime ideals of A we endow a spectral topology (in the sense of Grothendieck). We characterize irreducible closed subsets of Spec A and prove every irreducible closed subset of the space has a unique generic point. We give a sufficient condition for the space to be Noetherian. We study continuous maps between such spaces, and finally, we prove that Spec(Idl A) is a spectral space, where Idl A is the set of all ideals of A.
Paper Structure (4 sections, 26 theorems, 17 equations)

This paper contains 4 sections, 26 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spectra of skew braces
  4. Spectral topology

Key Result

Proposition 2.1

Suppose $A$ is a skew brace and $I$ is a normal subgroup of $\left(A,+\right)$. Then $I$ is an ideal of $A$ if and only if $A* I\subseteq I$ and $I* A\subseteq I$.

Theorems & Definitions (55)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • proof
  • ...and 45 more