Cancellation of projective modules over regular rings with comparability

AbstractThis paper answers, in the negative, Open Problem 4 in Goodearl's book on von Neumann regular rings: a directly finite regular ring R with s-comparability is constructed which is not unit-regular. It is shown, however, that the behaviour of these rings R is still quite good. They have cancellation of small projective modules and, in particular, their stable range is at most 2