BRAIDS, ORDERINGS AND ZERO DIVISORS

We begin with the observation that the group algebras CBn of Artin's b raid groups have no zero divisors or nontrivial units. This follows fr om 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 greater than or equal to 3 w hich is simultaneously left and right invariant. Nevertheless, we argu e that the subgroup of pure braids does possess a total ordering which is invariant on both sides. This follows from a general theorem regar ding orderability of certain residually nilpotent groups. As an applic ation, we show that the pure braid groups have no generalized torsion elements, although full braid groups do have such elements.