Semistar Dedekind Domains

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dc.creatorBaghdadi, Said El
dc.creatorFontana, Marco
dc.creatorPicozza, Giampaolo
dc.date2004-05-07
dc.date.accessioned2026-07-07T05:08:00Z
dc.date.available2026-07-07T05:08:00Z
dc.descriptionLet $D$ be an integral domain and $\star$ a semistar operation on $D$. As a generalization of the notion of Noetherian domains to the semistar setting, we say that $D$ is a $\star$--Noetherian domain if it has the ascending chain condition on the set of its quasi--$\star$--ideals. On the other hand, as an extension the notion of Prüfer domain (and of Prüfer $v$--multiplication domain), we say that $D$ is a Prüfer $\star$--multiplication domain (P$\star$MD, for short) if $D_M$ is a valuation domain, for each quasi--$\star_{_{f}}$--maximal ideal $M$ of $D$. Finally, recalling that a Dedekind domain is a Noetherian Prüfer domain, we define a $\star$--Dedekind domain to be an integral domain which is $\star$--Noetherian and a P$\star$MD. In the present paper, after a preliminary study of $\star$--Noetherian domains, we investigate the $\star$--Dedekind domains. We extend to the $\star$--Dedekind domains the main classical results and several characterizations proven for Dedekind domains. In particular, we obtain a characterization of a $\star$--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 $\star$--Dedekind domain if and only if the Nagata semistar domain Na$(D, \star)$ is a Dedekind domain. Several applications of the general results are given for special cases of the semistar operation $\star$.
dc.identifierhttps://arxiv.org/abs/math/0405124
dc.identifierhttp://arxiv.org/abs/math/0405124
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/71091
dc.subjectCommutative Algebra
dc.subjectAlgebraic Geometry
dc.subjectNumber Theory
dc.subject13A15; 13G05; 13E99
dc.titleSemistar Dedekind Domains
dc.typetext

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