Skew braces with no proper left ideals

TL;DR

This paper classifies finite left-simple skew braces, which are algebraic structures with no proper left ideals, and relates them to minimal Hopf--Galois structures on finite Galois extensions.

Contribution

It provides a partial classification of finite left-simple skew braces and connects these structures to minimal Hopf--Galois structures, expanding understanding of their algebraic properties.

Findings

Finite left-simple skew braces are characterized by having no proper left ideals.

Such skew braces correspond to minimal Hopf--Galois structures on finite Galois extensions.

The paper offers a partial classification of these algebraic structures.

Abstract

A skew brace $A = (A,\cdot,\circ)$ is said to be \textit{left-simple} if $A\neq1$ and it has no left ideal other than $1$ and $A$. The purpose of this paper is to give a partial classification of the finite left-simple skew braces. A result of Stefanello and Trappeniers implies that finite left-simple skew braces correspond precisely to minimal Hopf--Galois structures on finite Galois extensions of fields.