We prove that two arbitrary ideals $I \subset J$ in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of Specialization of Integral Dependence, which gives a condition for integral dependence in terms of the constancy of the multiplicity sequence in families.

中文翻译：

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重数序列和积分依赖

我们证明了等维和普遍悬链诺特局部环中的两个任意理想 $I \subset J$ 具有相同的积分闭包当且仅当它们具有相同的重数序列。我们还获得了积分依赖的特殊化原理，它根据族中的重数序列的恒定性给出了积分依赖的条件。