  We show that a strong form (the fully faithful version) of the generating
hypothesis, introduced by Freyd in algebraic topology, holds in the derived
category of a ring R if and only if R is von Neumann regular. This extends
results of the second author. We also characterize rings for which the original
form (the faithful version) of the generating hypothesis holds in the derived
category of R. These must be close to von Neumann regular in a precise sense,
and, given any of a number of finiteness hypotheses, must be von Neumann
regular. However, we construct an example of such a ring that is not von
Neumann regular, and therefore does not satisfy the strong form of the
generating hypothesis.
