Reduction Numbers and Balanced Ideals

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Let R be a Noetherian local ring and let I be an ideal in R. The ideal I is called balanced if the colon ideal J:I is independent of the choice of the minimal reduction J of I. Under suitable assumptions, Ulrich showed that I is balanced if and only if the reduction number, r(I), of I is at most the `expected' one, namely ℓ(I)- I+1, where ℓ(I) is the analytic spread of I. In this article we propose a generalization of balanced. We prove under suitable assumptions that if either R is one-dimensional or the associated graded ring of I is Cohen-Macaulay, then Jⁿ⁺¹:Iⁿ is independent of the choice of the minimal reduction J of I if and only if r(I) ≤ ℓ(I)- I+n.