A note on semiprime skew left braces and related semidirect products
Marco Castelli
TL;DR
The paper analyzes how semiprimeness and related primality notions behave under semidirect products of skew left braces, proving that if the product $B_1\\rtimes_{\\alpha} B_2$ is semiprime and $B_1$ is Artinian, then $B_1$ is semiprime, and that the semidirect product of strongly semiprime braces is strongly semiprime. It further develops the structure of semidirect products of simple braces to produce new strongly prime, non-simple skew left braces of abelian type, including a computer-free 576-element example and additional instances found with the GAP Ve24pack database. The work also provides counterexamples showing that semiprimeness of the product does not force semiprimeness of $B_2$, highlighting limits of transferring properties to factors. Collectively, these results extend ring-theoretic analogies to skew left braces, expand the catalog of strongly prime non-simple abelian-type braces, and present practical constructions supported by computational tools. These contributions deepen our understanding of how semiprime and strongly semiprime properties interact with semidirect product constructions in the theory of skew left braces.
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.