We define the radical and weight of a skew left brace and provide some basic properties of these notions. In particular, we obtain a Wedderburn type decomposition for Artinian skew left braces. Furthermore, we prove analogues of a theorem of Wiegold, a theorem of Schur and its converse in the context of skew left braces. Finally, we apply these results to detect torsion in the structure group of a finite bijective non-degenerate set-theoretic solution of the Yang–Baxter equation.

中文翻译：

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斜撑的径向和权重及其在Yang–Baxter方程解的结构组中的应用

我们定义了偏左括号的根部和权重，并提供了这些概念的一些基本属性。特别是，我们获得了Artinian偏斜左牙套的Wedderburn类型分解。此外，我们证明了Wiegold定理，Schur定理及其在左大括号歪斜情况下的逆定理的类似物。最后，我们将这些结果用于检测Yang–Baxter方程的有限双射非退化集合理论解的结构组中的扭转。