In this paper, we are primarily concerned with the behaviour of the centre with respect to the completion process for von Neumann regular rings at the pseudo-metric topology induced by a pseudo-rank function.
Let R be a (von Neumann) regular ring, and N a pseudo-rank function (all terms left undefined here may be found in ). Then N induces a pseudo-metric topology on R, and the completion of R at this pseudo-metric, , is a right and left self-injective regular ring. Let Z( ) denote the centre of whatever ring is in the brackets. We are interested in the map .
If R is simple, Z(R) is a field, so is discrete in the topology; yet Goodearl has constructed an example with Z(R) = R and Z(R) = C [5, 2.10]. There is thus no hope of a general density result.