From author’s introduction: “Let R be an integral domain with quotient field K. In Math. Z. 107, 301-306 (1968; Zbl 0167.036), J. Brewer showed that if R is not quasilocal, then R satisfies certain ring- theoretic properties if and only if \(R_ x=R[1/x]\) satisfies the corresponding property for each nonzero nonunit x in R. However, Brewer’s results do not hold for quasilocal integral domains.
In this paper, we extend this investigation and study quasilocal integral domains R with maximal ideal M such that R satisfies certain ring- theoretic properties for each nonzero x in M. It thus seems natural to study the overring \(R^{\#}=\{R_ x: x\) is a nonzero nonunit in \(R\}\) of R. We first show that \(R^{\#}\) may be characterized as the largest overring S of R such that \(R_ x=S_ x\) for each nonzero nonunit x in R. We relate \(R^{\#}\) to various ideal transforms. Let R be quasilocal with maximal ideal M. We show that \(R^{\#}=S(M)\), where S(M) is the S- transform introduced by Hays. In the special case in which M is finitely generated, \(R^{\#}=T(M)\), the usual ideal transform of Nagata. We also prove some general results about the S-transform and relate these to properties of R.”