The Ratliff-Rush ideal associated to a nonzero ideal I in a commutative Noetherian domain R with unity is I = ∞ n=1 (I n+1 : R I n) = {IS ∩ R : S ∈ B(I)}, where B(I) = {R[I/a] P : a ∈ I − 0, P ∈ Spec(R[I/a])} is the blowup of I. We observe that certain ideals are minimal or even unique in the class of ideals having the same associated Ratliff-Rush ideal. If (R, M) is local, quasi-unmixed, and analytically unramified, and if I is M-primary, then we show that the coefficient ideal I {k} of I, i.e., the largest ideal containing I whose Hilbert polynomial agrees with that of I in the highest k terms, is also contracted from a blowup B(I) (k) , which is obtained from B(I) by a process similar to "S 2-ification". This allows us to generalize the notion of coefficient ideals. We investigate these ideals in the specific context of a two-dimensional regular local ring, observing the interaction of these notions with the Zariski theory of complete ideals.
Abstract. Let R be a ¿-dimensional Noetherian quasi-unmixed local ring with maximal ideal M and an A/-primary ideal / with integral closure 7 . We prove that there exist unique largest ideals Ik for 1 < k < d lying between / and 7 such that the first k+\ Hilbert coefficients of / and Ik coincide. These coefficient ideals clarify some classical results related to 7. We determine their structure and immediately apply the structure theorem to study the associated primes of the associated graded ring of / .
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