Module braces: relations between the additive and the multiplicative groups

TL;DR

This paper introduces module braces ($R$-braces) generalizing previous concepts to arbitrary rings, explores their properties, and studies the relationship between their additive and multiplicative groups using algebraic methods.

Contribution

It generalizes the concept of braces to $R$-braces over any ring, providing new structural insights and explicit examples, and extends splitting and group relation results.

Findings

$R$-braces generalize previous braces to arbitrary rings.

Explicit examples of $R$-braces are constructed.

Under certain conditions, additive and multiplicative groups have the same element counts by order.

Abstract

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.