A note on semiprime skew left braces and related semidirect products

Abstract

In this paper, we focus on semiprime skew left braces provided by semidirect products. We show that if a semidirect product $B_1⋊B_2$ is semiprime and $B_1$ is Artinian, then $B_1$ must be semiprime. Moreover, we prove that the semidirect product of strongly semiprime skew left braces is strongly semiprime. Finally, following \cite[Question $1$]{smoktunowicz2024more}, we provide examples of skew left braces of abelian type that are non-simple and strongly prime.

12 pages, minor changes based on referee's comments. Accepted for publication in Bull. Math. Simon Stevin