USING GOING-UP TO CHARACTERIZE GOING-DOWN DOMAINS

A (commutative integral) domain R is called an AGU-domain if R subset of T satisfies the going-up property whenever T is an algebraic extension domain of R such that the natural map Spec(T) -> Spec(R) sends the maximal spectrum Max(T) onto Max(R). Any domain of (Krull) dimension 1 is an AGU-domain, as is any absolutely injective (ai-) domain. A quasilocal domain is an AGU-domain if and only if it is a going-down domain. A partial generalization is given for rings with nontrivial zero-divisors. An example is given of a two-dimensional Prufer (hence going-down) domain with exactly two maximal ideals which is not an AGU-domain.