On products of abelian skew braces

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.