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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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