Semistar Dedekind Domains

Let $D$ 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 Pr"ufer domain (and of Pr"{u}fer $v$–multiplication domain), we say that $D$ is a Pr"ufer $\star$–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 Pr"{u}fer 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 $\star$.