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.Comment: 12 page
