Abstract :
Let B = (b1, ..., bg) R subset of or equal to I be ideals in a Noetherian ring R, let F = F(R, B) be the form ring of R with respect to B, and let G = F/IF. Then the main results in this paper give several characterizations of when b1,..., bg are I-independent, three of which are: G = (R/I)[X1, ..., Xg]; height(I) ≥ g and QG is primary for all primary ideals Q in R/I; and, (QBn + m : fm(b1, ..., bg)R) ∩ Bn = QBn for all primary ideals Q in R that contain I, for all forms fm(X1, ..., Xg) of degree m that have at least one coefficient not in Rad(Q), and for all nonnegative integers m and n. It is then shown that if these conditions hold and if C and D are ideals in R that contain I, then the following hold for all n ≥ 0: (C ∩ D)In = CIn ∩ DIn; if CBn subset of or equal to DBn, then C subset of or equal to D; and, Ass(R/C) subset of or equal to Ass(R/CBn) subset of or equal to Ass(R/C) union or logical sum Ass(R/Bn). In particular, these characterizations and results hold when I = B, so this special case yields several new characterizations of when the form ring F(R, B) is the polynomial ring (R/B)[X1, ..., Xg], among which is that the images in RM of b1 = bg form an RM-sequence for all maximal ideals M in R that contain B. Finally, the characterizations and results also hold when I = Ba, the integral closure of B, and this special case yields an asymptotic sequence version of the R-sequence results.
