CENTRAL NILPOTENCY OF LEFT SKEW BRACES AND SOLUTIONS OF THE YANG–BAXTER EQUATION

This paper delves into the study of centrally nilpotent skew braces. In particular, we study their torsion theory, we introduce an “index” for subbraces, but we also show that the product of centrally nilpotent ideals need not be centrally nilpotent. To cope with these examples, we introduce a special type of nilpotent ideal, using which, we define a good Fitting ideal. Also, a Frattini ideal is defined and its relationship with the Fitting ideal is investigated. A key ingredient is the characterisation of the commutator of ideals in terms of star products, which solves a problem of Bonatto and Jedlička (J. Algebra Appl. 22:12 (2023), art. id. 2350255). Moreover, we provide an example showing that the idealiser of a subbrace does not exist in general.