  In this paper we prove that the classical Lie bracket of vector fields can be
generalized to the noncommutative setting by antisymmetrizing (in a suitable
noncommutative sense) their compositions. To accomplish this, we study linear
differential operators on modules over a unital associative algebra equipped
with an exterior algebra. We provide necessary and sufficient conditions for
jet modules to be representing objects for differential operators, which is, a
priori, a key desideratum for any theory of noncommutative differential
geometry. Further, we construct natural symbol maps, which play a crucial
r\^ole in the construction of the aforementioned Lie bracket.
