# Factorizations of skew braces

## TL;DR

This paper introduces the concept of strong left ideals in skew braces, demonstrating their role in decomposing solutions to the Yang-Baxter equation and exploring their structural properties.

## Contribution

It develops the theory of strong left ideals in skew braces, including factorization and analogs of Itô's theorem, with applications to the Yang-Baxter equation.

## Key findings

- Strong left ideals enable non-trivial decompositions of set-theoretic solutions.
- Analogues of Itô's theorem are established for skew left braces.
- Classification of skew braces with no non-trivial proper ideals is provided.

## Abstract

We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of It\^{o}'s theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang-Baxter equation. Finally, we classify skew braces that contain no non-trivial proper ideals.