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Strongly divisorial ideals and complete integral closure of an integral domain. (English) Zbl 0596.13002
Let D be a commutative integral domain with unity. An ideal I of D is called strongly divisorial if \(I=II^{-1}=(I^{-1})^{-1}\) [cf. J. Querré, Bull. Sci., II. Sér. 95, 341-354 (1971; Zbl 0219.13015)]. The main results of this paper are: \((i)\quad I\quad is\) strongly divisorial if and only if \(I=D:R\) for some overring R of D and the map \(R\to D:R\) gives a one-to-one correspondence between the set of overrings of D of the form \(I^{-1}\) for some ideal I of D and the set of strongly divisorial ideals of D; \((ii)\quad D^*=\cup \{I^{-1};\quad I\quad is\quad strongly\quad divisorial\},\) where \(D^*\) is the complete integral closure of D, i.e. the set of all elements x of the quotient field of D such that there exists an element \(a\neq 0\) of D with \(a^ nx\in D\) for some positive integer n. These results can be applied to study Mori domains which are defined by the property that every increasing sequence of integral divisorial ideals is stationary [cf. N. Raillard, C. R. Acad. Sci., Paris, Sér. A 280, 1571-1573 (1975; Zbl 0307.13010)] and ”normal” domains which are defined by the property \((D:D^*)\neq 0\) [cf. J. Querré, op. cit.].
Reviewer: Ngo Viet Trung

MSC:
13A05 Divisibility and factorizations in commutative rings
13B02 Extension theory of commutative rings
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