  We introduce the notion of a braid group parametrized by a ring, which is
defined by generators and relations and based on the geometric idea of painted
braids. We show that the parametrized braid group is isomorphic to the
semi-direct product of the Steinberg group (of the ring) with the classical
braid group. The technical heart of the proof is the Pure Braid Lemma, which
asserts that certain elements of the parametrized braid group commute with the
pure braid group.
  More generally, we define, for any crystallographic root system, a braid
group and a parametrized braid group with parameters in a commutative ring. The
parametrized braid group is expected to be isomorphic to the semi-direct
product of the corresponding Steinberg group with the braid group. The first
part of the paper (described above) treats the case of the root system $A_n$;
in the second part, we handle the root system {$D_n$}. Other cases will be
treated in the sequel.
