A note on semiprime skew left braces and related semidirect products

TL;DR

This paper investigates the properties of semiprime skew left braces formed via semidirect products, establishing conditions under which components are semiprime and providing examples of non-simple, strongly prime braces.

Contribution

It proves that semiprime semidirect products with Artinian components have semiprime factors and that strongly semiprime braces preserve their property under semidirect products, also providing new examples.

Findings

Semiprime semidirect products with Artinian components have semiprime factors

Semidirect product of strongly semiprime braces is strongly semiprime

Constructed examples of non-simple, strongly prime braces of abelian type

Abstract

In this paper, we focus on semiprime skew left braces provided by semidirect products. We show that if a semidirect product $B_1\rtimes B_2$ 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.