On universally going-down underrings. II (Q1124639)

[For part I see ibid. (to appear; Zbl 0665.13006).] This paper gives a new characterization of an often-characterized class of integral domains. Let R be an integral domain with quotient field K. It is proved that the inclusion map \(A\to B\) satisfies universally going down [in the sense of the author and \textit{M. Fontana}, J. Algebra 90, 410-429 (1984; Zbl 0544.13004)] for each pair \(A\subset B\) of subrings of R having quotient field K if and only if either R is isomorphic to an overring of \({\mathbb{Z}}\) or \(R=K\) is an algebraic field extension of a finite field. This result sharpens a theorem of part I of this paper. The proof uses a result of the author about the analogous situation involving going-down [from Bull. Aust. Math. Soc. 36, 503-513 (1987; Zbl 0619.13004)], as well as standard facts about weak/semi-normalization and pullbacks.