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On minimal overrings. (Sur les suranneaux minimaux.) (French) Zbl 0972.13007

Definition: Let \(R\) and \(T\) be integral domains, \(T\) containing \(R\) strictly. If \(R\) is not a field and if the extension \(R\subset T\) does not admit proper intermediate rings then one says that \(T\) is a minimal overring of \(R\) and the extension is minimal.
J. Sato and K.-I. Yoshida showed that under certain conditions a noetherian domain may not have a minimal overring. – In the present paper, the authors first determine the minimal overrings of a Dedekind domain \(R\) as those \(T_{(\eta)}= \bigcap\{R_p,p\) maximal ideal of \(R,p\neq \eta\}\), where \(\eta\) is a fixed maximal ideal of \(R\). Moreover:
Theorem. If \((R, \eta)\) is a reflexive local domain and \(\eta\) is not principal, then
(1) \(R\) has a minimal overring \(T\) entire over \(R\);
(2) \(T\) is local and \(T/R\) is a simple \(R\)-module.
Theorem. For a minimal extension \(R\subset T\) the following holds:
(1) \(T\) is \(R\)-flat if and only if \(R\) is integral closed in \(T\);
(2) \(T\) is never strongly \(R\)-flat;
(3) If \(T\) is \(R\)-flat, then \(\overline T\) is \(\overline R\)-flat.
Theorem. Let \(T\) be a minimal overring of \(R\) such that \(T\) is \(R\)-flat.
(1) If \((R:T)=0\) then an ideal \(Q\subset T\) is maximal (respectively, prime) iff \(Q\cap R\) is maximal (respectively, prime) in \(R\);
(2) If a is local then \(T\) is local.
Theorem. If \(T\) is \(R\)-flat and \((R:T)=0\) then \(\dim R=\dim T\).
Theorem. Let \(R\) be a minimal extension. If \(R\) is a Macaulay ring then \(T\) is also Macaulay.

MSC:

13B02 Extension theory of commutative rings
13G05 Integral domains
13C14 Cohen-Macaulay modules
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