The main theorem gives a
necessary and sufficient condition for each Rees locality ℒ = R[tb,u](M,tb,u) of a local
ring (R,M) with respect to a principal ideal bR in R to be either an Hi-ring (that is,
for all prime ideals p in ℒ such that height p = i, depth p = altitude ℒ− i) or a
homogeneously Hi-ring (same condition holds for homogeneous p). Numerous
corollaries follow concerning the cases: R is complete; R is Henselian; and, ℒ is Hi,
for all i ≧ 0. A generalization to ideals generated by more than one element is
given, and we relate the results to two of the chain conjectures on prime
ideals.
