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Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation. (English) Zbl 1473.16028

Radicals are important tools in ring theory and they also have been found new applications to the theory of the Yang-Baxter equation.
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, L. Guarnieri and 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 radical of a skew left brace, that is the intersection of all of its maximal ideals, and the 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.

MSC:

16T25 Yang-Baxter equations
16N80 General radicals and associative rings
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References:

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