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.1991 AMS Mathematics subject classification. 13A15. Keywords and phrases. semistar operations, localizing systems, valuation domains.

A class of integer-valued functions defined on the set of ideals of an integral domain [Formula: see text] 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 [Formula: see text]-operation. We use this class of functions to recover familiar semistar operations such as the [Formula: see text]-operation and to give a solution to a conjecture by Chapman and Glaz when the ring is a valuation domain.