标题： 关于阿贝尔斜 braces 的积 显示英文标题

标题： On products of abelian skew braces

摘要： 本文的主要目的是研究斜左brace通过阿贝尔子brace的分解。 我们证明了关于两个阿贝尔群之积的经典伊藤定理的一个斜brace理论类比：如果斜brace$B = A_1A_2$是两个阿贝尔斜子brace$A_1$和$A_2$的乘积，并且$A_1$是$B$的左、右理想，则$[B, B]^B$作为$B$的换位理想是一个阿贝尔brace。 如果 $A_1$ 是 $B$ 的左理想（不必是双边理想），我们证明存在一个强左理想包含于 $B$ 中且包含于 $A_1$ 或 $A_2$。 我们还证明了分解环的有关理想的分解，这些理想是交换子环的和与积。 显示英文摘要

摘要： The main objective of this paper is to study factorisations of skew left braces through abelian subbraces. We prove a skew brace theoretical analog of the classical It\^o's theorem about product of two abelian groups: if $B = A_1A_2$ is a skew brace which is the product of two abelian skew subbraces $A_1$ and $A_2$, and $A_1$ is a left and right ideal of $B$, then the commutator ideal $[B, B]^B$ of $B$ is an abelian brace. If $A_1$ is a left (non-necessarily right) ideal of $B$, we show that there exists a strong left ideal of $B$ contained in $A_1$ or $A_2$. We also show factorisations of relevant ideals of factorised braces that are sums and products of abelian subbraces.