On Going Down For Simple Overrings II

This paper deals with two themes involving the going down (GD) behavior of the overrings of a domain R. First, as illustrated by several results in 3] and 4], hypotheses about GD often replace stronger assumptions about flatness or one-dimensionality in sets of conditions implying that R is Prüfer. Second, as shown by the two-dimensional example in 4]. Remarks (iii)], some sort of finiteness condition is required in conjunction with GD hypotheses in order for R to be Prüfer.