  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.
