Domains integral over each underring (Q1107578)

An underring of a commutative integral domain is a subring with the same quotient field. Domains with their underrings seminormal were characterized by \textit{D. E. Dobbs} and \textit{T. Ishikawa} in Tokyo J. Math. 10, 157-159 (1987; Zbl 0629.13002). It is shown here, in particular, that apart from domains integral over either \({\mathbb{Z}}\) or a finite field, the domains R of the title are those for which \(char(R)>0\) and either just one valuation ring of K, the quotient field of R, does not contain R or, equivalently, \(R\subset V\) for all valuation rings V of K such that \(V\cap R\) is an underring of R.