  We settle in this paper a question left open in our paper ``Modular Hecke
algebras and their Hopf symmetry'', by showing how to extend the Rankin-Cohen
brackets from modular forms to modular Hecke algebras. More generally, our
procedure yields such brackets on any associative algebra endowed with an
action of the Hopf algebra of transverse geometry in codimension one, such that
the derivation corresponding to the Schwarzian derivative is inner. Moreover,
we show in full generality that these Rankin-Cohen brackets give rise to
associative deformations.
