Semistar Dedekind Domains
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 Prüfer domain (and of Prüfer
v
--multiplication domain), we say that
D
is a Prüfer
⋆
--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 Prüfer domain, we define a
⋆
--Dedekind domain to be an integral domain which is
⋆
--Noetherian and a P
⋆
MD. 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
⋆
.