AI 中文总结
本文证明斜括号中双侧性、$\star$-积结合性与$A\star A$中心性三者中任意两者蕴含第三者，并通过根环给出结合$\star$-积的双侧斜括号的纤维积描述、正合序列及幂零指数界，同时分类相关结构。
AI 中文摘要
对于括号，$\star$-积的结合性等价于双侧性。我们将此结果推广到斜括号，证明以下三个条件中任意两个蕴含第三个：双侧性、$\star$-积的结合性以及$A\star A$在加法群中的中心性。然后，我们通过与其加法阿贝尔化的根环相关联，研究具有结合$\star$-积的双侧斜括号。这产生了模一个零化理想的典范纤维积描述、将高阶$\star$-积与根环的幂联系起来的正合序列、左、右和强$\star$-级数的一致性，以及其幂零指数的尖锐界。我们还获得了下中心列的显式描述，分类了加法阿贝尔化为无限循环的结构，并刻画了它们何时是双斜的。
英文摘要
For braces, associativity of the $\star$-product is equivalent to two-sidedness. We extend this result to skew braces by proving that any two of the following conditions imply the third: two-sidedness, associativity of the $\star$-product, and centrality of $A\star A$ in the additive group. We then study two-sided skew braces with associative $\star$-product through the radical ring associated with their additive abelianization. This yields a canonical fibre-product description modulo an annihilator ideal, exact sequences relating higher $\star$-products to the powers of the radical ring, coincidence of the left, right and strong $\star$-series, and sharp bounds on their nilpotency indices. We also obtain an explicit description of the lower central series, classify the structures whose additive abelianization is infinite cyclic, and characterize when they are bi-skew.
Comments17 pages
