On products of abelian skew braces

TL;DR

This paper explores the structure of skew braces, establishing an analog of Itô's theorem for abelian skew braces and analyzing factorization properties involving ideals and subbraces.

Contribution

It introduces a skew brace analog of Itô's theorem and investigates factorization and ideal properties in abelian skew braces.

Findings

If a skew brace is a product of two abelian skew subbraces with certain ideal conditions, its commutator ideal is abelian.

Existence of strong left ideals within factorized braces under specific conditions.

Factorizations of ideals as sums and products of abelian subbraces.

Abstract

The main objective of this paper is to study factorisations of skew left braces through abelian subbraces. We prove a skew brace theoretical analog of the classical It\^o's theorem about product of two abelian groups: if $B = A_1A_2$ is a skew brace which is the product of two abelian skew subbraces $A_1$ and $A_2$, and $A_1$ is a left and right ideal of $B$, then the commutator ideal $[B, B]^B$ of $B$ is an abelian brace. If $A_1$ is a left (non-necessarily right) ideal of $B$, we show that there exists a strong left ideal of $B$ contained in $A_1$ or $A_2$. We also show factorisations of relevant ideals of factorised braces that are sums and products of abelian subbraces.