  Let N be a finitely generated module over a Noetherian local ring (R,m). We
give criteria for the height of the order ideal N^*(x) of an element x \in N to
be bounded by the rank of N. The Generalized Principal Ideal Theorem of Bruns,
Eisenbud and Evans says that this inequality always holds if x \in mN, but if x
is a minimal generator the question is more subtle; for example, when R is
graded and N represents a vector bundle on Proj R, the question is connected
with the vanishing of the top chern class. We show that the inequality does
hold if the hypothesis x \in mN becomes true after first extending scalars to
some local domain and then factoring out torsion. We also give conditions under
which it holds in terms of residual intersections and integral closures of
modules.
  We derive information about order ideals that leads to bounds on the heights
of trace ideals of modules--even in circumstances where we do not have the
expected bounds for the heights of the order ideals.
