Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation (Q2032942)

Radicals are important tools in ring theory and they also have been found new applications to the theory of the Yang-Baxter equation. \textit{W. Rump} [J. Algebra 307, No. 1, 153--170 (2007; Zbl 1115.16022)] proved that any radical ring produces an involutive non-degenerate solution; more generally, radical rings correspond to two-sided braces, i.e., left braces that also are right braces. In this context, \textit{L. Guarnieri} and \textit{L. Vendramin} [Math. Comput. 86, No. 307, 2519--2534 (2017; Zbl 1371.16037)] introduced a generalization of left braces, namely skew left braces, to study non-involutive bijective solutions. In the paper under review, they are introduced two important tools: the \textit{radical} of a skew left brace, that is the intersection of all of its maximal ideals, and the \textit{weight} of a skew left brace as the minimal number of generators needed to generate it as an ideal. These two notions are essential to prove several brace-theoretic analogues of classical theorems in ring theory and group theory. For instance, in analogy to the Artin-Wedderburn decomposition theorem for semisimple rings, it is shown that the quotient of an Artinian skew left brace by its radical is a product of simple skew left braces. Furthermore, according to Wiegold's thoerem, it is proved that each Artinian perfect skew left brace has weight equal to one. Besides, a theorem of Schur and its converse are provided in the context of skew left braces. All the result are used to study the torsion of the structure group of a set-theoretic solution and several examples are given.