A class of integer-valued functions defined on the set of ideals of an integral domain R is investigated. We show that this class of functions, which we call ideal valuations, are in one-to-one correspondence with countable descending chains of finite type, stable semistar operations with largest element equal to the e-operation. We use this class of functions to recover familiar semistar operations such as the w-operation and to give a solution to a conjecture by Chapman and Glaz when the ring is a valuation domain.

中文翻译：

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半星运营的下降链

定义在整数域理想集上的一类整数值函数R被调查。我们表明，我们称之为理想估值的这类函数与有限类型的可数降链一一对应，稳定半星运算的最大元素等于e-手术。我们使用这类函数来恢复熟悉的半星运算，例如w-操作并在环是估值域时给出查普曼和格拉兹猜想的解决方案。