论文标题
模块括号：添加剂和乘法组之间的关系
Module braces: relations between the additive and the multiplicative groups
论文作者
论文摘要
在本文中，我们定义了一类括号，我们称其为模块牙套或$ r $ braces，它们是添加剂组在环$ r $上也具有模块结构的牙套，而伽马功能的值是$ r $ $ $ - 模型的自动形态。在文献中已经考虑了这类牙套，如果环$ r $是一个领域：我们将定义推广到任何环$ r $，以与支架相关的所谓伽马功能进行重新诠释，并证明这类括号享受所有自然物业所需的所有自然属性。我们展示了R型的明确例子，并研究了与环$ r $分裂相关的模块牙套的分裂，从而概括了Byott的结果，即用Nilpotent多能多样化组的支架作为其Sylow子组的总和。 本文的核心是在最后两个部分中，其中，使用来自交换代数和数字理论的方法，我们研究了$ r $ - 布拉斯的加性和乘法组之间的关系，表明添加群的某些分解为\ emph {small}（在某种意义上取决于$ r $），然后添加元素相同的序列和乘积数量相同的序列，这些数字是相同的。在某些情况下，此结果大大扩大了本期已知的结果的应用范围。
In this paper we define a class of braces, that we call module braces or $R$-braces, which are braces for which the additive group has also a module structure over a ring $R$, and for which the values of the gamma functions are automorphisms of $R$-modules. This class of braces has already been considered in the literature in the case where the ring $R$ is a field: we generalise the definition to any ring $R$, reinterpreting it in terms of the so-called gamma function associated to the brace, and prove that this class of braces enjoys all the natural properties one can require. We exhibit explicit example of R-braces, and we study the splitting of a module braces in relation to the splitting of the ring $R$, generalising thereby Byott's result on the splitting of a brace with nilpotent multiplicative group as a sum of its Sylow subgroups. The core of the paper is in the last two sections, in which, using methods from commutative algebra and number theory, we study the relations between the additive and the multiplicative groups of an $R$-brace showing that if a certain decomposition of the additive group is \emph{small} (in some sense which depends on $R$), then the additive and the multiplicative groups have the same number of element of each order. In some cases, this result considerably broadens the range of applications of the results already known on this issue.
