Oukessou, M.; Miri, A. On minimal overrings. (Sur les suranneaux minimaux.) (French) Zbl 0972.13007 Extr. Math. 14, No. 3, 333-347 (1999). 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. Reviewer: G.-E.Winkler (Berlin) Cited in 1 Review MSC: 13B02 Extension theory of commutative rings 13G05 Integral domains 13C14 Cohen-Macaulay modules Keywords:minimal overrings PDFBibTeX XMLCite \textit{M. Oukessou} and \textit{A. Miri}, Extr. Math. 14, No. 3, 333--347 (1999; Zbl 0972.13007) Full Text: EuDML