Spectral topologies on skew braces

TL;DR

This paper introduces a spectral topology on the set of prime ideals of skew braces, characterizes its properties, and explores the topological structure of these spaces, including conditions for Noetherianity and spectrality.

Contribution

It defines a new spectral topology on prime ideals of skew braces and analyzes its topological properties, including irreducibility, generic points, and spectral space conditions.

Findings

Irreducible closed subsets have unique generic points

The space Spec A can be Noetherian under certain conditions

Spec(Idl A) is shown to be a spectral space

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.