  If $S$ is a discrete semigroup, then $\beta S$ has a natural,
left-topological semigroup structure extending $S$. Under some very mild
conditions, $U(S)$, the set of uniform ultrafilters on $S$, is a two-sided
ideal of $\beta S$, and therefore contains all of its minimal left ideals and
minimal idempotents. We find some very general conditions under which $U(S)$
contains prime minimal left ideals and left-maximal idempotents.
  If $S$ is countable, then $U(S) = S^*$, and a special case of our main
theorem is that if a countable discrete semigroup $S$ is a weakly cancellative
and left-cancellative, then $S^*$ contains prime minimal left ideals and
left-maximal idempotents. We will provide examples of weakly cancellative
semigroups where these conclusions fail, thus showing that this result is
sharp.
