Let R R be a d d -dimensional Noetherian quasi-unmixed local ring with maximal ideal M M and an M M -primary ideal I I with integral closure I ¯ \overline I . We prove that there exist unique largest ideals I k {I_k} for 1 ≤ k ≤ d 1 \leq k \leq d lying between I I and I ¯ \overline I such that the first k + 1 k + 1 Hilbert coefficients of I I and I k {I_k} coincide. These coefficient ideals clarify some classical results related to I ¯ \overline I . We determine their structure and immediately apply the structure theorem to study the associated primes of the associated graded ring of I I .