Going-down pairs of commutative rings
An extension \(R\subseteq T\) of (commutative unitary) rings satisfies going-down (GD), if given prime ideals \(P_0\subset P\) in \(R\) and \(Q\) in \(T\) with \(Q\cap R=P\), there exists a prime ideal \(Q_0\) in \(T\) satisfying \(Q_0\subset Q\) and \(Q_0\cap R=P_0\). And a ring \(R\) is called GD-ring, if \(R/P\subset T\) satisfies GD for every prime ideal \(P\) of \(R\) and every domain \(T\) containing \(R/P\) [cf. \textit{D. E. Dobbs}, Houston J. Math. 23, 1-12 (1997; Zbl 0896.13006)]. In the paper under review, the author studies ring extensions \(R\subset T\), such that every intermediate ring extension \(R\subset A\subset B\subset T\) (respectively every intermediate ring \(R\subset A\subset T\)) satisfies GD (respectively is a GD-ring). For example, he shows that the paranthetical condition above holds for \(R\subset T\) with \(R\) zero-dimensional iff for each minimal prime ideal \(P\) of \(T\) the transcendence degree of \(T/P\) over \(R/P\cap R\) is \(\leq 1\). Related conditions, obtained replacing GD by a similar property are also studied, e.g. INC, open map spectra, finite fiber [INC means the ``incomparability property'' described by \textit{T. J. Papick}, Trans. Am. Math. Soc. 219, 1-37 (1976; Zbl 0345.13005); p. 28]. The paper contains numerous examples.
scientific article; zbMATH DE number 1884331
scientific article