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Zbl 0685.13009
Picavet-l'Hermitte, Martine
(l'Hermitte, Martine; Hermitte, Martine)
Ordres de Gorenstein. (Gorenstein orders). (French)
[J] Ann. Sci. Univ. Blaise Pascal Clermont-Ferrand II 91, Math. 24, 1-32 (1987). ISSN 0249-7042

An integral order A over a Dedekind ring Z is a finite extension of Z such that the quotient field K of A is an algebraic separable extension of the quotient field Q of Z. The aim of this paper is to give criterions for an integral order to be a Gorenstein ring. To do this the author studies the complement and the different of A. Let Tr denote the trace map of K over Q. For each ideal I of A, the complement $I\sp*$ is the set of elements $x\in K$ with Tr(xI)$\subset Z$. The different D(A) is the ideal $A:A\sp*$. Using the relationship between Gorenstein rings and canonical ideals [see {\it J. Herzog} and {\it E. Kunz}, ``Der Kanonische Modul eines Cohen-Macaulay-Rings'', Lect. Notes Math. 238 (1971; Zbl 0231.13009) the author shows that A is a Gorenstein ring if and only if $A\sp*$ or D(A) is invertible. There are also numerical characterizations of Gorenstein integral orders in terms of the notion of the index of two fractional ideals [which is due to {\it A. Fröhlich} and {\it J. W. S. Cassels} (see Algebraic number theory, Proc. Instruct. Conf. Lond. Math. Soc., NATO Adv. Study Inst. (1967; Zbl 0153.074)].
[Ngo Viet Trung]

MSC 2000:: *13H10 Special types of local rings
13F05 Dedekind and Pruefer rings and their generalizations
16H05 Orders and arithmetic, separable associative algebras

Keywords: integral order; Dedekind ring; Gorenstein ring; complement; different; canonical ideals; fractional ideals

Citations: Zbl 0231.13009; Zbl 0153.074

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