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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 modulestructure over a ring $R$, and for which the values of the gamma functions areautomorphisms of $R$-modules. This class of braces has already been consideredin the literature in the case where the ring $R$ is a field: we generalise thedefinition to any ring $R$, reinterpreting it in terms of the so-called gammafunction associated to the brace, and prove that this class of braces enjoysall the natural properties one can require. We exhibit explicit example ofR-braces, and we study the splitting of a module braces in relation to thesplitting of the ring $R$, generalising thereby Byott's result on the splittingof 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 methodsfrom commutative algebra and number theory, we study the relations between theadditive and the multiplicative groups of an $R$-brace showing that if acertain decomposition of the additive group is \emph{small} (in some sensewhich depends on $R$), then the additive and the multiplicative groups have thesame number of element of each order. In some cases, this result considerablybroadens the range of applications of the results already known on this issue.

Module braces: relations between the additive and the multiplicative groups