We begin with the observation that the group algebras [Formula: see text] of Artin's braid groups have no zero divisors or nontrivial units. This follows from the recent discovery of Dehornoy that braids can be totally ordered by a relation < which is invariant under left multiplication. We then show that there is no ordering of B n , n ≥ 3 which is simultaneously left and right invariant. Nevertheless, we argue that the subgroup of pure braids does possess a total ordering which is invariant on both sides. This follows from a general theorem regarding orderability of certain residually nilpotent groups. As an application, we show that the pure braid groups have no generalized torsion elements, although full braid groups do have such elements.