Skew braces with no proper left ideals

A skew brace $A=(A,\cdot ,\circ )$ is said to be $\mathit{\text{left-simple}}$ if $A\mathrm{\ne }1$ and it has no left ideal other than $1$ and $A$. The purpose of this paper is to give a partial classification of the finite left-simple skew braces. A result of Stefanello and Trappeniers implies that finite left-simple skew braces correspond precisely to minimal Hopf–Galois structures on finite Galois extensions of fields.