Semistar Dedekind domains

By S. E. Baghdadi, Marco Fontana, G. Picozza. 2004

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Abstract Let D be an integral domain and ★ a semistar operation on D . As a generalization of the notion of Noetherian domains to the semistar setting, we say that D is a ★-Noetherian domain if it has the ascending chain condition on the set of its quasi-★-ideals. On the other hand, as an extension the notion of Prufer domain (and of Prufer v -multiplication domain), we say that D is a Prufer ★-multiplication domain (P★MD, for short) if D M is a valuation domain, for each quasi-★ f -maximal ideal M of D . Finally, recalling that a Dedekind domain is a Noetherian Prufer domain, we define a ★-Dedekind domain to be an integral domain which is ★-Noetherian and a P★MD. For the identity semistar operation d , this definition coincides with that of the usual Dedekind domains and when the semistar operation is the v -operation, this notion gives rise to Krull domains. Moreover, Mori domains not strongly Mori are ★-Dedekind for a suitable spectral semistar operation. Examples show that ★-Dedekind domains are not necessarily integrally closed nor one-dimensional, although they mimic various aspects, varying according to the choice of ★, of the “classical” Dedekind domains. In any case, a ★-Dedekind domain is an integral domain D having a Krull overring T (canonically associated to D and ★) such that the semistar operation ★ is essentially “univocally associated” to the v -operation on T . In the present paper, after a preliminary study of ★-Noetherian domains, we investigate the ★-Dedekind domains. We extend to the ★-Dedekind domains the main classical results and several characterizations proven for Dedekind domains. In particular, we obtain a characterization of a ★-Dedekind domain by a property of decomposition of any semistar ideal into a “semistar product” of prime ideals. Moreover, we show that an integral domain D is a ★-Dedekind domain if and only if the Nagata semistar domain Na( D ,★) is a Dedekind domain. Several applications of the general results are given for special cases of the semistar operation ★.

Subfields: Algebra and Number Theory, Organic Chemistry

View Paper: https://www.semanticscholar.org/paper/29659250dc9d50221b324c64299e482616a49986