On the structure of some nilpotent braces

Abstract

We prove a criteria for nilpotency of left braces in terms of the $\star$-central series and also discuss Noetherian braces, obtaining some of their elementary properties. We also show that if a finitely generated brace $A$ is Smoktunowicz-nilpotent, then the additive and multiplicative groups of $A$ are likewise finitely generated.