Going-down underrings

Abstract

Let R be an integral domain with quotient field K. Ten conditions equivalent to “Either R is algebraic over ℤ or t.d.(R/Fp) ≤ 1 for some p” are given. One of these conditions, referred to in the title, is “Each extension of subrings of R having quotient field K satisfies the going-down property.” As consequences, other classes of rings are also characterised.