  We derive the Hu-Paz-Zhang master equation for a Brownian particle linearly
coupled to a bath of harmonic oscillators on the plane with spatial
noncommutativity. The results obtained are exact to all orders in the
noncommutative parameter. As a by-product we derive some miscellaneous results
such as the equilibrium Wigner distribution for the reservoir of noncommutative
oscillators, the weak coupling limit of the master equation and a set of
sufficient conditions for strict purity decrease of the Brownian particle.
Finally, we consider a high-temperature Ohmic model and obtain an estimate for
the time scale of the transition from noncommutative to ordinary quantum
mechanics. This scale is considerably smaller than the decoherence scale.
