Well-centered overrings of a commutative ring in pullbacks and trivial extensions (Q422017)

Let \(R\) be a commutative ring. By \(\mathrm{Tot}(R)\) we mean the total quotient ring of \(R\), i.e. the localization of \(R\) at non-zerodivisors. A subring \(S\) of \(\mathrm{Tot}(R)\) containing \(R\) is said to be an overring of \(R\). \textit{W. Heinzer} and \textit{M. Roitman} [J. Algebra 272, No. 2, 435--455 (2004; Zbl 1040.13002)] defined and investigated the well-centered overrings of an integral domain. An overring \(S\) of an integral domain \(R\) is said to be well-centered over \(R\) if for every \(b\in S\), there exists unit \(u\in S\) such that \(ub\in R\). In the paper under review the authors extend this definition to an arbitrary commutative ring \(R\) and then study the condition under which a ring \(S\) is an overring of the trivial extension \(R\ltimes E\) of \(R\) by an \(T(R)\)-module \(E\) and also the condition under which \(S\ltimes M\) is well-centered over \(R\ltimes M\) where \(R\subseteq S\) is an extension of rings and \(M\) is an \(S\)-module. Then, in connection with some questions left open in [Heinzer and Roitman, loc. cit.], the authors give an example of a non-coherent ring \(R\) such that every finitely generated well-centered overring of \(R\) is a localization of \(R\) and every flat overring of \(R\) is well-centered on \(R\).NEWLINENEWLINEThe authors also deal with well-centered overrings of a pullback domain \(R\) issued from a diagram. In particular, they give examples of a Noetherian, and also a non-Noetherian, pseudo-valuation domain \(R\) such that each overring is well-centered on \(R\) (\textit{E. Bastida} and \textit{R. Gilmer} [Mich. Math. J. 20, 79--95 (1973; Zbl 0239.13001)] characterizes pseudo-valuation domains in terms of pullbacks).