On two-sided skew braces

TL;DR

This paper introduces weakly trivial skew braces, classifies them, and demonstrates their importance in understanding two-sided skew braces, leading to new results and unifying existing concepts in the field.

Contribution

It provides a classification of weakly trivial skew braces and shows their role in the structure of two-sided skew braces, also unifying prime and semi-prime notions.

Findings

Weakly trivial skew braces are classified and shown to be fundamental.

New results relating additive and multiplicative groups of two-sided skew braces.

Prime and semi-prime notions coincide for two-sided skew braces.

Abstract

In order to study two-sided skew braces, we introduce the notion of weakly trivial skew braces. We give a classification of such skew braces and show that they are essential in the study of two-sided skew braces. As an application, we obtain new and generalize known results relating the additive and multiplicative group of two-sided skew braces. Further, we show that two a priori different notions of prime and semi-prime skew braces, as introduced by Konovalov, Smoktunowicz and Vendramin, coincide for two-sided skew braces.