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Some classification of certain integral domains via conductor overrings and semistar operations. (English) Zbl 1454.13005

The main goal of the present paper is to classify integral domains \(R\) in terms of their conductor overrings (i.e., overrings of the type \((I:I)\) where \(I\) is a fractional ideal of \(R\)) and valuation overrings, and to apply it to classify integral domains that possess certain distinguished semistar-operational properties.
Following this line, the first half of this article is focused on overring-theoretical characterizations of domains whose integral closure is a valuation domain. For instance, among several other results, the author proves that an integral domain \(R\) is a PVD (i.e., pseudo-valuation domain, after Hedstrom-Houston) whose integral closure is a valuation domain if and only if it is a seminormal domain whose valuation overrings are prime conductor overrings (i.e., overrings of the type \((P:P)\) where \(P\) is a prime ideal of \(R\)). Using these results, the author provides a characterization of certain classes of totally divisorial domains (i.e., domains whose overrings are all divisorial).
Another application of the previous results is used, in the second part of the paper, to study new properties of semistar operations (after Okabe and Matsuda) and, in particular, the behavior of the composition of semistar operations (after Picozza). The author investigates when the composition of two semistar operations of an integral domain \(R\) is a semistar operation on \(R\). In particular, he studies integral domains \(R\) where every composition of a pair of semistar operations on \(R\) is again a semistar operation on \(R\). Specifically, the author proves that when such \(R\) is integrally closed (respectively, stable), then the set of semistar operations on \(R\) is totally ordered and \(R\) is a valuation domain (respectively, totally divisorial conducive domain).
This article, derived in part from the author’s dissertation, is very well organized and written in an accurate self contained manner and can be used as a solid starting point for new investigations in multiplicative ideal theory.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
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